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 Multiplicative programming theory and algorithms
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 Boger, George
 Publication Date:
 1999
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 English
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 vii, 137 leaves : ; 29 cm.
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 Algorithms ( jstor )
Approximation ( jstor ) Efficiency objectives ( jstor ) Heuristics ( jstor ) Linear programming ( jstor ) Mathematics ( jstor ) Objective functions ( jstor ) Optimal solutions ( jstor ) Polyhedrons ( jstor ) Polytopes ( jstor ) Decision and Information Sciences thesis, Ph. D ( lcsh ) Dissertations, Academic  Decision and Information Sciences  UF ( lcsh )
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 Thesis (Ph. D.)University of Florida, 1999.
 Bibliography:
 Includes bibliographical references (leaves 131136).
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 Printout.
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 Vita.
 Statement of Responsibility:
 by George Boger.
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MULTIPLICATIVE PROGRAMMING: THEORY AND ALGORITHMS
By
GEORGE BOGER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
ACKNOWLEDGMENTS
I would like to thank my entire supervisory committee Dr. Harold Benson, Dr.
Selcuk Erenguc, Dr. Asoo Vakharia, and Dr. Richard Francis for their time and helpful
feedback on my dissertation. I am especially grateful to my committee chairman, Dr.
Benson, for suggesting the topic of multiplicative programming problems and for his
tremendous assistance and unending support. Without his help, this dissertation would
not have been completed. I would also like to thank Mr. Erijang Sun for proving some
theoretical results needed to support my dissertation topic.
I am also grateful to the DIS department chairman, Dr. Erenguc, for providing an
assistantship and for allowing me to teach undergraduate courses during my time at the
University of Florida. The teaching experience was an enjoyable and rewarding
experience.
I would like to thank my family for their encouragement and emotional support. I
would also like to thank my colleagues in the Ph.D. program for their friendship and their
support.
Finally, I am in debt to my master's degree advisor, Dr. Frederick Buoni, at the
Florida Institute of Technology, for his guidance. He suggested multiple objective linear
programming as a topic for my thesis. While working on the thesis, I met Dr. Benson
during a visit to FIT to present a talk related to multiple objective linear programming.
Dr. Benson agreed to serve on my master's degree committee and later recruited me for
the DIS Ph.D. program.
TABLE OF CONTENTS
page
ACKNOW LEDGM ENTS.............................................................................
ABSTRACT........................... ........................... ................
CHAPTERS
1 INTRODUCTION ............................................................. .......
1.1. The Multiplicative Programming Problem............................. ........... 1
1.2. Reformulations of the Multiplicative Programming Problem.....................4
1.3. Purpose and Organization of the Dissertation...........................................
2 A REVIEW OF THE LITERATURE ON MULTIPLICATIVE
PROGRAMMING PROBLEMS .......................... ........................9
2.1. Organization of the Literature Review ................................... ............. 9
2.2. Methods to Solve Problems (LMP2), (GLMP), and (CLMP)................... 13
2.2.1. Methods Based on Quadratic Programming ................................. 15
2.2.2. Methods Based on Searching the Outcome Set ............................ 17
2.2.3. Methods Based on Solving a Parametric Master Problem.............. 22
2.2.4. Methods Based on Polyhedral Annexation................................... 28
2.3. Extensions of Algorithms for Problem (LMP2) to Solve Problem
(LM P) when p 3............................. ...........................................32
2.4. Methods to Solve Problems (CMP), (GCMP) and (CCMP) ...................32
2.4.1. Methods Based on Solving a Reformulated Problem ...................33
2.4.2. A Method Based on Outer Approximation .....................................37
2.5. Methods to Solve Problem (LMP) as a Concave Minimization
Problem ............................ ... ..... ...................... 38
3 CONCAVE MULTIPLICATIVE PROGRAMMING PROBLEMS:
ANALYSIS AND AN EFFICIENT POINT SEARCH HEURISTIC
FOR THE LINEAR CASE ......................... ..............................40
3.1. Introduction.................. ......................... .. .......................... 40
3.2. A analysis ........................................ ................................................4 1
3.3. Efficient Point Search Heuristic ......................... .........................52
3.4. Computational Results............................. ...........................62
3.5. D iscussion........................................ .............................................. 69
4 A GENERAL MULTIPLICATIVE PROGRAMMING PROBLEM IN
OUTCOMESPACE ............................ .... ........................71
4.1. Introduction.......................................................... .......................... 7 1
4.2. Results for the General Case of Problem (Py,) ......................................73
4.3. Results for Convex and Polyhedral Cases of Problem (P ) ...................78
4.4. D iscussion............................................ ................................................ 96
5 AN OUTCOMESPACE CUTTINGPLANE ALGORITHM FOR
LINEAR MULTIPLICATIVE PROGRAMMING ................................98
5.1. Introduction...................................... ............................. .............. 98
5.2. Theoretical Prerequisites ....................................... .............. 100
5.3. OutcomeSpace, CuttingPlane Algorithm............................................ 104
5.3.1. Strict Local Optimal Solution Search.......................................... 105
5.3.2. Cutting Plane Construction. ................................................. 107
5.3.3. Termination Test. ................................... ........................... 109
5.3.4. OutcomeSpace, CuttingPlane Algorithm ................................. 110
5.4. Implementation.......................... ...... ............................ 114
5.5. Exam ple .......................................................... ............................. 119
5.6. Concluding Remarks ........................................ ........... ..... 124
6 SUMMARY AND FUTURE RESEARCH................................................. 125
6.1. Introduction.................................................. .......................... 125
6.2. Future Research on the Heuristic Algorithm......................................... 125
6.3. Future Research on an Global Solution Algorithms................................ 127
R E FE R E N C E S .......................................................................... ................................ 131
BIOGRAPHICAL SKETCH................................................ ............. 137
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MULTIPLICATIVE PROGRAMMING: THEORY AND ALGORITHMS
By
George Boger
December 1999
Chairman: Harold P. Benson
Major Department: Decision and Information Sciences
Multiplicative programming problems are mathematical optimization problems in
which the objective function contains a product of several real valued functions defined
over a common domain and the feasible decisions are described by a nonempty set. These
optimization problems have some important applications in engineering, finance,
economics, and other fields. Multiplicative programming problems, however, are difficult
global optimization problems that are known to be NPhard.
This dissertation has two purposes. The first is to develop and test a heuristic
algorithm that finds a good solution, though not necessarily a globally optimal solution,
for the linear multiplicative programming problem. The second purpose is to develop a
global solution algorithm for the linear multiplicative programming problem that is
potentially more efficient than existing algorithms for this problem.
To evaluate the effectiveness in practice of the heuristic algorithm, we have
written a FORTRAN computer program and used it to solve 260 randomly generated
linear multiplicative programming problems of various sizes. Our experimental results
show that the computational requirements of the heuristic algorithm are not overly
burdensome when compared to the effort required to solve a linear multiplicative
programming problem.
The framework of the outcomespace, cuttingplane algorithm is taken from a
pure cutting plane, decision setbased method developed by Horst and Tuy for solving
concave minimization problems. By adapting the approach of this method to an outcome
space reformulation of the linear multiplicative programming problem, rather than
directly applying the method to the original decision set formulation, it is expected that
considerable computational savings can potentially be obtained. We also show how
additional computational benefits might be obtained by implementing the new algorithm
appropriately. To illustrate the new algorithm, we apply it to the solution of a sample
problem.
CHAPTER 1
INTRODUCTION
1.1. The Multiplicative Programming Problem
Multiplicative programming problems are mathematical optimization problems in
which the objective function contains a product of several real valued functions defined
over a common domain and the feasible decisions are describe by a nonempty set. These
problems occur is a wide variety of application areas.
For example, Konno and Inori (1989) studied a bond portfolio optimization
problem in which the portfolio's performance is measured by a number of indices such as
the average coupon rate, the average terminal yield, and the average length to maturity.
The goal of the portfolio manager is to improve the performance of the portfolio by
purchasing or selling bonds in the marketplace subject to some limiting constraints. The
manager must consider multiple incomparable objectives such as maximizing the average
terminal yield and minimizing the average maturity time. Konno and Inori choose to
optimize several objectives simultaneously by multiplying them together since the
objectives do not share a common scale.
Another example of a multiplicative programming problem, given in Maling,
Mueller and Heller (1982), is a packaging problem encountered in designing very large
scale integrated circuit (VLSI) chips and laying out building floor plans or manufacturing
plant facilities. In the problem, the overall rectangular dimensions of the feasible layout
2
plans are constrained rather than fixed. Different layout plans with differing overall
rectangular dimensions are obtained according to how the components of a system are
arranged within each plan. The objective is to find the arrangement of components that
minimizes the overall layout area subject to certain constraints on the area and the
perimeter of the layout.
Henderson and Quandt (1971, p. 15) also give an application of multiplicative
programming problems. Their example is from microeconomics. In their example, a
rational consumer wishes to find a combination of two commodities to purchase from
which he will derive the highest possible level of satisfaction. Budgetary constraints and
the availability of the commodities limit the quantities the consumer may purchase. The
consumer's level of satisfaction is captured by his utility function, which is assumed to be
the product of the quantities of the two commodities. The rational consumer's problem is
then formulated as maximizing his utility function subject to the budgetary and
commodity availability constraints.
The multiplicative programming problem or, more briefly, the multiplicative
program, may be formulated mathematically as
(Px) minh(x)= (f,(x), s.t. xe X,
where p > 2 is an integer, X c R", and, for each j = 1, 2,..., p, f : X R satisfies
f,(x)2 0 for all x X. For simplicity we will assume throughout this proposal that the
minimum of problem (Px) is achieved at some point x' e X. In addition we will assume
that p is significantly less than n since this holds for virtually all applications of
multiplicative programming problems. If f, (I) = 0 for some j E {l, 2,..., p} and some
eE X, then clearly is a global optimal solution. This condition can be checked by
solving p minimization problems min {f (x)xe X j = 1, 2,...,p. Therefore, we may
assume without loss of generality that, for each j = 1, 2,..., p, f (x)> 0 holds for all
xe X.
The objective function h of problem (Px) is generally not a convex function. As
a result, problem (Px) belongs to a class of nonconvex programming problems called
global optimization problems. In contrast to convex programming problems, there may be
many local minima for problem (Px) that are not globally optimal. Conventional local
optimization methods based on gradients, subgradients, conjugate directions, or the
KarushKuhnTucker conditions, for instance, are at best guaranteed only to find a local
minimum. These methods must then terminate, since there is neither a local criterion for
certifying the global optimality of a given solution nor a way to determine how to proceed
to a better solution if the solution is not globally optimal. From the perspective of
computational complexity, problem (Px) is a difficult problem that is known to be NP
hard even when the objective function is simply h(x)= x, x2 and the feasible region X
is a polyhedron (Matsui 1996).
When in addition to the assumptions given previously for problem (Px), X is a
convex set and, for each j= 1, 2,..., p, f : X R is a concave function, we obtain the
concave case of problem (Px), called the concave multiplicative programming problem.
The convex case of problem (Px), called the convex multiplicative programming
problem, is obtained when, in addition to the assumptions made previously for problem
(Px), X is a convex set and, for each j = 1, 2,..., p, f, : X R is a convex function.
A special linear case of problem (Px), called the linear multiplicative programming
problem, is obtained when, in addition to the assumptions make previously for problem
(Px), X is a compact polyhedron and, for each j = 1, 2,..., p, f : X R is a linear
function (Konno and Kuno 1992).
1.2. Reformulations of the Multiplicative Programming Problem
During the 1990's there has been a resurgence of interest in problem (Px).
Encouraged by the rapid advances in high speed computing, researchers began developing
and testing new methods for solving global optimization problems that arise in practical
applications, including problem (Px).
Included among the global optimization methods used to solve problem (Px) for
the special case when p = 2 are various parametric simplex methodbased algorithms
(e.g., Konno and Kuno 1992, Konno and Kuno 1995, Konno, Yajima, and Matsui 1991,
and Schaible and Sodini 1995), branch and bound procedures (e.g., Kuno 1996 and Muu
and Tam 1992), and various other types of algorithms (e.g., Konno and Kuno 1990,
Pardalos 1990, and Tuy and Tam 1992).
When p > 2, globally solving problem (Px) has been shown empirically to
require considerably more computational effort than when p = 2 (see, e.g., Ryoo and
Sahinidis 1996). A smaller number of the algorithms for solving problem (Px) when
5
p > 2 solve the problem directly without reformulating it as an outcomespace problem.
Included among these, for instance, is the polyhedral annexation algorithm of Tuy (1991).
Most of the algorithms for solving problem (Px) when p > 2, however, solve the
problem indirectly by globally solving an outcomespace reformulation of the problem
instead. This is because in practical applications p is routinely much smaller than n,
often by two or more orders of magnitude. As a result, working in R' is computationally
less challenging than working in R".
Let y E RP denote the pvector with jth entry equal to y,, j = 1, 2,..., p. For
each j= 1, 2..., p, let e, R satisfy
j >sup fj(x) s.t.xe X,
where j = +oo is possible, and let ye RP denote the vector with jth entry equal to 9,,
j = 1, 2,..., p. Let f(x) denote the vector f(x)= f,(x) f2(xl...,fp(x)]', where
fj: X  R, j = 1, 2,..., p, are the functions used in defining problem (Px). Thoai
(1991) and later Konno and Kuno (1995) based their outer approximation algorithms for
respectively solving the convex and linear cases of problem (Px) on one of the more
direct reformulations of problem (Px) as an outcomespace problem. Their reformulation
is given by
(P',) min yj, s.t. y Y',
j=1
where
Y ={ye R'jf(x)
Falk and Palocsay (1994) based their branch and bound, image space algorithm
for the linear case of problem (Px) on another outcomespace reformulation that is
closely related to problem (Py,). Their reformulation is given by
(Pr) minly,, s.t. ye Y,
j=I
where
Y= ye RP y=Cx forsomexe X}
and C is a (pxn) matrix whose rows are cT, j = 1, 2,..., p.
1.3. Purpose and Organization of the Dissertation
This dissertation has two main purposes. The first is to develop and test a heuristic
algorithm that finds a good solution, though not necessarily a globally optimal solution,
for the linear case of problem (Px). The second purpose is to develop an exact global
solution algorithm for the linear case of problem (Px) that is potentially more efficient
than existing algorithms for this problem.
Since the linear multiplicative programming problem is known to be an NPhard,
multiextremal global optimization problem, it is inherently more difficult to globally
solve than a convex programming problem of the same size. In some application cases, a
solution will adequately meet the requirements of a user; see, e.g., Konno and Inori
(1989). In these cases, the use of a heuristic algorithm seems to be appropriate for finding
a satisfactory solution. To date, however, there is no known heuristic algorithm tailored to
finding a good solution for the linear multiplicative programming problem. In their
review of algorithms for solving problem (Px), Konno and Kuno (1995) do not mention
7
any heuristic algorithms for problem (Px), and our survey of the literature has revealed
none.
To develop the heuristic algorithm, we first analyze the concave multiplicative
programming problem. The analysis yields a new way to write a concave multiplicative
programming problem as a concave minimization problem. As a result, a concave
multiplicative programming problem can be solved by using any existing concave
minimization algorithm without resorting to a reformulation of the problem. We also
show that some relationships exist between concave multiplicative programming
problems and certain multipleobjective mathematical programs. These relationships are
exploited to develop the heuristic algorithm for the linear case of problem (Px).
For cases where a linear multiplicative program must be solved for an exact global
optimal solution, we expect that globally solving the outcomespace reformulation (Py)
instead will result in a significant decrease in the computational effort over that required
to directly solve the problem. This is because in typical applications of linear
multiplicative programs, p is several orders of magnitude smaller than n. As a result,
working in RP should be computationally less challenging than working in R".
To globally solve the outcomespace reformulation (P,,) of a linear multiplicative
program, we develop an outcomespace, pure cutting plane algorithm that works in R'.
The framework for the algorithm is taken from a pure cutting plane, decision setbased
concave minimization method developed by Horst and Tuy (1993). We show how to
adapt this method to solving the reformulation (P,,) of a linear multiplicative program
for a global extreme point optimal solution. Once this global solution is found, we can
recover a globally optimal solution for the linear multiplicative program in decision
space. As a further computational enhancement, we also show that for purposes of
implementation, the mechanics of the outcomespace, cuttingplane algorithm can be
applied to the smaller problem (Pr) instead of problem (P,,).
The organization of the proposal is as follows. In Chapter 2 we present a review
of the literature on multiplicative programming problems. In Chapter 3 we analyze the
concave multiplicative programming problem, apply the results to develop a heuristic
algorithm for the linear multiplicative programming problem, and report test results using
the heuristic algorithm on some randomlygenerated problems. In Chapter 4 we analyze
the reformulation problem (P,,) and show that, under certain convexity assumptions on
Y, problem (P,) has a global extreme point optimal solution y* e Y. We then present
a procedure that is guaranteed to find a strict local optimal extreme point solution for the
reformulation problem (P, ) of the linear multiplicative program. In Chapter 5 we
present an outcomespace, cuttingplane algorithm for globally solving a linear
multiplicative program. The algorithm employs the strict local optimal search procedure
presented in Chapter 4. We also illustrate the algorithm by applying it to the solution of a
sample problem. Finally, in Chapter 6, we give an overall summary and conclusions, and
we discuss directions for further research.
CHAPTER 2
A REVIEW OF THE LITERATURE ON MULTIPLICATIVE PROGRAMMING
PROBLEMS
2.1. Organization of the Literature Review
In this chapter we present a review of the literature on methods proposed for
solving multiplicative programming problems. The only known literature review on
multiplicative programming problems appears in Konno and Kuno (1995). In their
literature review Konno and Kuno defined multiplicative programming problems as "a
class of minimization problems containing a product of several convex functions either in
its objective function or in its constraints." They included problems in which the
objective function contained the summation of a convex function and the product of
convex functions.
Konno and Kuno (1995) organized their literature review based on whether the
problem data are linear or nonlinear and on the number of functions that appear in the
objective function. They considered solution methods for the following multiplicative
programming problems.
The first multiplicative programming problem considered by Konno and Kuno is
the special case of quadratic programming
(LMP2) min f (x)= ((c,x)+ d)((c2,x)+d2), s.t. x D,
where D := xe R' Ax > b, x > 0O is a nonempty polytope (bounded polyhedron) in
10
which A is an mxn matrix, be R", and, for each i= 1, 2, c' e R" \{0} and d, e R.In
addition, it is assumed that, for each x e D, (c', x) + d, > 0, i= 1, 2.
The second multiplicative programming problem that they considered is the
convex multiplicative programming problem
(CMP) minf(x)=I f (x, s.t.xe X,
where X C R' is a nonempty, compact, convex set and, for each j = 1, 2,..., p,
f: R"  R is a convex function that satisfies f (x)> 0 for all xe X.
Konno and Kuno (1995) considered two special cases of problem (CMP): (1) the
case where p = 2 and (2) the case where p 2 and the problem data are linear. The
second case may be defined as the following extension of problem (LMP2):
(LMP) min f (x) = c',x)+di], s.t.xe D,
where p 2 is an integer and, for each i = 1, 2,..., p, (c',x) + d > 0 holds for all xe D.
Finally, Konno and Kuno (1995) considered three classes of problems related to
problem (CMP). In the first class is the following problem:
(GCMP) min f(x) = fo(x)+ fj (x)f42,(x s.t.xe X,
j=1
where, for each j = 0,1,...2q, f, : R R is a convex function that satisfies f (x)> 0
for all xe X.
The second class is a special case of (GCMP) in which q = 1 and the problem
data are linear. This class may be defined as the following extension of problem (LMP2):
(GLMP) min f(x) = (c, x) + (( x) + d,)((c, x) + d2, s.t. xe D,
where co R" and c', di, i = 1, 2, and D are defined as in problem (LMP2).
The third class of problems considered by Konno and Kuno (1995) is the
minimization of a convex function over a feasible region that includes a product of
convex functions in its constraint set.
Konno and Kuno's coverage of the literature is not exhaustive. They focused on
algorithms that have been demonstrated by computational experiments to be practical for
reasonably large problems (Konno and Kuno 1995, p. 370). Algorithms proposed by
Konno, Kuno, and their associates have been tested on randomly generated problems and
the results reported. However, computational results have not been reported by most of
the other researchers and therefore their methods were not included in the review.
Since the publication of the review by Konno and Kuno, two more multiplicative
programming problems have been discussed in the literature. The first problem adds a
convex function to the objective of problem (LMP2) to obtain the problem:
(CLMP) min f(x)= g(x)+((c',x) +dX(c2,x)+ d2), s.t.x D,
where g : R" R is a twice differentiable convex function and c', d,, i= 1,2, and D
are defined as in problem (LMP2). The second problem adds a convex function to
problem (CMP) to obtain the problem:
(CCMP) min f(x)= fo(x)+ f f(x), s.t.xe X,
where f : R"  R is a convex function that satisfies fo(x)> 0 for all xe X and f,,
j = 1, 2,..., p, and X are defined as in problem (CMP).
The emphasis of this review will be on optimization problems in which a product
of functions appears in the objective function. Optimization problems with objective
functions that are comprised of a summation of a function and the product of functions
are also included in the review. Methods proposed for solving these problems may be
adapted to solve a problem whose objective function is strictly a product of functions by
setting the added function to the null function. The functions that appear in the objective
function will be either convex or linear functions since to date these are the only
multiplicative programming problems to appear in the literature. In this review we will
not consider optimization problems in which a product of functions appears in the
constraint set.
Like the review of Konno and Kuno (1995), this literature review is organized
based on whether the problem data are linear or nonlinear and on the number of functions
that appear in the objective function. It is divided into the following four sections. Section
2.2 reviews the methods proposed to solve problems (LMP2), (GLMP), and (CLMP).
Section 2.3 reviews the methods to solve problem (LMP) that are extensions of methods
for problem (LMP2). Section 2.4 reviews the methods to solve problems (CMP),
(GCMP), and (CCMP). Section 2.5 reviews the methods to solve problem (LMP) as a
concave minimization problem.
The rationale for organizing the literature review in this way is as follows.
Historically, the first algorithms for solving multiplicative programming problems were
specifically proposed for solving problem (LMP2). Problems (GLMP) and (CLMP) are
grouped with problem (LMP2) since they were conceived as extensions of that problem.
Several of the algorithms proposed for solving problem (LMP2) can be extended to solve
the problem (LMP), since they do not depend upon having only two functions in the
product term of the objective function. Problems (LMP2), (LMP), (GLMP), and (CLMP)
contain linear functions and polyhedral feasible regions. Algorithms for solving these
problems are implemented with the aid of the simplex method, which is used to solve
linear programming subproblems. The problems (CMP), (GCMP), and (CCMP) contain
nonlinear data and must rely on other optimization methods to solve nonlinear convex
programming problems. The latter three problems are therefore placed in a separate
group. Problems (GCMP) and (CCMP) are included in the group with problem (CMP)
because only one article addresses each problem, and they were conceived as extensions
of problem (CMP). Finally, two articles appeared in the literature that proposed solving
problem (LMP) as a concave minimization problem using techniques that the authors had
previously developed.
Table 2.1 gives a summary of the multiplicative programming problems
considered in this literature review along with the assumptions placed on the feasible
region and the objective function of each problem.
2.2. Methods to Solve Problems (LMP2), (GLMP), and (CLMP)
The methods for solving problem (LMP2), (GLMP), and (CLMP) are further
divided into four categories. In the first category are those methods that analyze problem
(LMP2) as a special case of quadratic programming. In the second category are
algorithms that analyze problem (LMP2) by searching the outcome set. In the third
category are the algorithms that solve an easier parametric programming problem rather
than directly solving problems (LMP2), (GLMP), and (CLMP). In the last category are
Table 2.1. Summary of Multiplicative Program Types and Assumptions on Problems
n 1.1 Assumptions on the
Problem Feasible Ri on Objective Function Assumptions on the Objective Function
LMP2 D is abounded polyhedron. ((c',x)+ d)((c,x)+ d2) (c',x)+d, > i = 1, 2,forall xe D.
GLMP D is a bounded polyhedron. (co,x)+((cl,x)+d,)((c ,x)+d2) (cox)>Oand(c,x)+d >0, i=1,2,forall
xe D.
g : R" R is a twice differentiable convex
CLMP D is a bounded polyhedron, g(x)+((cl,x)+dX(c2,x)+d2) function and (c',x)+ d, > 0, i=1, 2, for all
xe D.
LMP D is a bounded polyhedron. f c',x) +d (c',x)+d > O, i= 2,..., p, forall xeD.
For each j= 1, 2, ...,p, f: R"' R isa
CMP X is a compact convex set. fj (x) convex function that satisfies f (x)> 0 for all
xe X.
Foreach j=0,1, ...,p, f :R" 4R isa
GCMP X is a compact convex set. fo (x) + f (ix)f2i (x) convex function that satisfies fj (x)> 0 for all
j=1
xe X.
Foreach j = 0,1, ...,p, f : R" + R isa
CCMP X is a compact convex set. f (x) + f (x) convex function that satisfies f (x)> 0 for all
X _X.
two algorithms that solve problem (LMP2) based on the method of polyhedral
annexation.
2.2.1. Methods Based on Quadratic Programming
Since the objective function of problem (LMP2) can be expressed as
f(x)= ((c'x) + d,)((c,x) + d)= I xTQx+ rrx+ dd,,
where re R', and Q is a real symmetric n xn matrix, problem (LMP2) is a special class
of quadratic programming. Swarup (1966a and 1966b) was the first researcher to analyze
problem (LMP2) in this way, but he did not propose any exact solution algorithms. His
two articles are included in the literature review for completeness. Pardalos (1990) also
analyzed problem (LMP2) in this way, and he proposed an exact global solution
algorithm.
Swarup (1966a) showed that if both linear functions (c',x) + d,, i = 1, 2 are
positive over the feasible region D, the objective function f is quasiconcave over D, It
is well known that generally for any local minimizer of a quasiconcave function over a
polytope, there exists an extreme point local minimizer over the polytope that has the
same function value. Swamp proposed a simplex based method for finding such a local
optimal solution. The key to the algorithm is a test that determines if entering a given
nonbasic variable into the current simplex basis will lower the objective function value. A
simplex basis of a local optimal solution can be reached by beginning at any feasible
basis and moving through a sequence of simplex tableaux by pivoting in qualifying
nonbasic variables until none remain. Once a local optimal solution is found, the
16
algorithm stops. No information is available to either certify the global optimality of the
solution or to determine how to proceed to an improved solution.
In another work, Swamp (1966b) formulated the following parametric linear
program by introducing an auxiliary variable 4 and moving one of the linear functions
into the constraint set:
(MP1) minF(x;4) =(c',x)+d,
s.t. x eD,
(c',x)+ d, 2= 5 0.
Since (c2, x) + d2 appears in the constraint set, dual pricing information is
available to determine the value of (c',x)+ d, as 4 is set to achievable values of
(c2, x)+ d2 over D. Swamp derived a test that uses this information to determine when
4 is set to a level that corresponds to a local optimal solution. All local optimal solutions
can then theoretically be found by parametrically solving problem (MPI) over all
achievable values of 4. A global optimal solution x' of problem (LMP2) can then be
found by identifying a global solution (x', 4') of problem (MP1).
Pardalos (1990) observed that if c' and c2 are linearly independent, then the
Hessian matrix Q of the objective function of problem (LMP2) has one positive
eigenvalue and one negative eigenvalue, and the remaining eigenvalues are equal to zero.
By applying the spectral decomposition theorem of linear algebra, the objective function
can be rewritten in terms of two variables. The problem can then be solved by examining
the vertices of an orthogonal projection of the feasible region D into a twodimensional
17
polytope in the space of the two variables used in the rewritten objective function.
Pardalos (1990) proposed an algorithm that enumerates all vertices of the two
dimensional polytope until an optimal vertex is found. The algorithm may require an
exponential number of steps, but its average computational time complexity is bounded
by a polynomial.
2.2.2. Methods Based on Searching the Outcome Set
The objective function of problem (LMP2) can be expressed as the composite
yy(p) of two mappings, where, for each xe R", p(x)= ((cl,x+d,,(c2,x)+d), and, for
each y e R2, YI(y) = Y Y2 The mapping ip maps each point x e D into a point
y =(y,, y2) where y,:=(c',x)+d, andy2 :=(c2,x)+d2. Since y, and y are linear
functions, ( is a linear transformation and hence the linear structure of D is preserved
(Rockafellar 1970). The image of D under (p is then the compact, convex polyhedron
Y:= {y R2y, =(c',x )+d,, y, = (c2,x) + d2 for some x D
called the outcome polyhedron. A global optimal solution of problem (LMP2) can be
found by finding a point of Y that globally minimizes the product y, y2. Since the
search is conducted in Y e R2 rather than R", it may be possible to economize on the
computational effort required to solve problem (LMP2).
Three articles, Aneja, Aggarwal, and Nair (1984), Falk and Palocsay (1994), and
Thoai (1991) proposed algorithms for solving problem (LMP2) based on searching the
outcome set using outer approximation techniques. Outer approximation is a global
optimization technique that uses a decreasing sequence of simple sets to approximate the
feasible region. The approximations are used in a series of optimization problems that are
easier to solve than the original problem. These optimization problems are sequentially
solved until a global optimal solution to the original problem is found. The technique has
been very useful in solving global optimization problems in which the feasible region Z
is a polytope and the global optimal solution is known to be an extreme point of Z. In
this form of outer approximation, the algorithm begins by finding a simple polytope
P0 D Z with an easily defined inequality representation and an easily calculated set of
vertices. A series of algorithmic iterations follows that builds a sequence of decreasing
polytopes P, D P z D : Z in which one polytope is generated in each iteration. In an
iteration k of the algorithm, the original objective function is evaluated at the extreme
points of Pk to find an optimal solution vk. If vk is an extreme point of Z, then v" is a
global optimal solution to the original problem. Otherwise, a portion of Pk \ Z is cut off
to form P,,,. The point v* is part of the region cut off; i.e., vk is not included in the
polytope P+,. The cut is made by adding a constraint called a cutting plane constraint to
the constraint set that defines P The cutting plane constraint adds additional vertices to
Pk.+ that were not present in P. and therefore they must be calculated.
Aneja, Aggarwal, and Nair (1984) proposed an algorithm that examines the
solutions associated with the bicriterion programming problem:
(BCP) VMIN(y,=(c',x)+d,,y2=(c,x)+d2),
s.t. xe D.
The intent of problem (BCP) is to simultaneously minimize the two criterion
functions y, and y2. Conflicts usually exist between the two criterion functions that
prevent a single point of D from simultaneously minimizing both functions. The usual
notion of an optimal solution used in single objective linear programming is replaced by
the concept of efficient solutions when discussing the solutions of problem (BCP). A
solution x is an efficient solution of problem (BCP) if YE D and, whenever for each
i=1,2, (c',x)+di <(c',5)+d, forsome x D,then (c',x)+d, =(c', )+d,. i=l, 2.
The set of efficient points of D is mapped by Vp into a set of points on the surface of Y
called the efficient frontier.
Aneja, Aggarwal, and Nair (1984) showed that a global optimal solution of
problem (LMP2) is attained at an efficient extreme point x' of D that is mapped by ip
into an extreme point (y,', y) in the efficient frontier of Y. Their algorithm searches the
efficient frontier for an extreme point that minimizes yy,2 by using a modified outer
approximation technique. Initially the legs of a rightangle triangle form the first
approximation of the efficient frontier. The "rise" and the "run" values of the slope of the
hypotenuse are two positive scalar values. The functions y, = (c',x) +d, and
y= (c, x) + d, are multiplied by these values and then summed to form a single linear
objective function. This objective function is then minimized over the feasible region D.
It is well known that the minimizer i of such a linear program is an efficient extreme
point of D (Steuer 1986). The solution to the linear program finds another point (9,, 92)
on the efficient frontier that is used to subdivide the initial triangle into two triangles. The
20
algorithm is then repeated using each of the smaller triangles. The algorithm terminates
when there are no more extreme points of the efficient frontier that need to be searched.
In the algorithm of Aneja, Aggarwal, and Nair (1984), a new vertex must be
calculated for each triangle. This is easily done by solving two systems of two equations
in the unknowns y, and y2. This special technique however, can not be easily extended
to handle cases where p > 2.
Falk and Palocsay (1994) also proposed a solution algorithm that searches among
the extreme points of Y using a modified outer approximation technique. In the first
phase of the algorithm, the two linear programs
1, = min(c',x + d and 12 = minc2, x + d,
.xD / xD 
are solved for optimal solutions x' and x2 respectively. Two initial vertices y' and y2
of Y are then
y' =((c',x')+d,,(c2,x)+d2) and y2 =((c',x2)+d,,(c2,x)+d).
An initial polytope in outcomespace containing an optimal solution for the problem
(YP) min y
is y, /1, and y2 /12 and an inequality a,y, +ay2 <1, where a, and a2 are determined
such that ay, +a2y2 = 1 passes through the point
y = argmin t'i(y:, y2), (y, y2
i=1.2
In each iteration of the algorithm, values for a, and a2 are updated and a linear program
of the form
21
(YLP) min ay, + a2y2
is solved to remove portions of the initial polytope from the search for an optimal
solution for problem (YP). The new vertices generated at each iteration are easily
calculated since the isovalue contours of problem (YLP) are linear. The algorithm
terminates when the optimal value of problem (YLP) is one.
The algorithm proposed by Thoai (1991) for solving problem (LMP2) uses an
outer approximation technique that begins by enclosing the outcome set Y in a rectangle
P,. In an iteration k of the algorithm, the extreme point (v,, v) of the outer
approximation that yields the lowest value of the product y, y, is found. A linear
program is then used to determine if the extreme point (,, 2 ) maps to a feasible point i
of D. If not, information is obtained from the linear program to generate a cutting plane
constraint that slices off the extreme point (,, 2 ) from the polytope P, The new
vertices generated by the cut are then calculated using a conventional approach (see
Horst, Pardalos, and Thoai 1995 or Horst and Tuy 1993). Since the method of
determining these new vertices is not dependent on the fact that the dimension of the
outcome set is two, Thoai's algorithm can be extended to handle cases where p > 2.
In the algorithms of Aneja, Aggarwal, and Nair (1984) and Thoai (1991), the only
variations in the linear programs used in successive iterations involve changes in
objective function coefficients. The authors gain some computational efficiency by
restarting the simplex method at the optimal solution of the previous iteration. Only a few
simplex pivots are then generally needed to produce a new optimal solution.
22
2.2.3. Methods Based on Solving a Parametric Master Problem
The difficulty in solving problem (LMP2) is caused by the product form of the
objective function. Konno and Kuno (1992) added a parameter 4 and formed the
following problem that they called the master problem:
(MP2) minF(x;4)= ((c',x +d,)+ (( +d2),
s.t. xeD, 5 >0.
Notice that for a fixed value 4' of 4, problem (MP2) is a linear programming
problem. To solve problem (MP2), Konno and Kuno proposed using a parametric
objective function simplex method to the find critical values of 4 at which new bases
become optimal. The values of the objective function F are then evaluated at these
bases. A global optimal solution (x', ') of problem (MP2) is found by choosing the
basis that minimizes F over these values. Konno and Kuno (1992) showed that if
(x', 4') is an optimal solution of problem (MP2), then x is a global optimal solution of
problem (LMP2).
Konno and Kuno tested this algorithm on randomly generated problems (LMP2)
with nonnegative problem data that ranged in size from (m, n) = (30, 50) to (220, 200).
Their computational experiments showed that the amount of computational time needed
to solve problem (LMP2) is not much different from that required to solve linear
programs of the same size.
In Konno and Kuno (1995) the authors slightly simplified the above parametric
method by redefining the auxiliary parameter so that convex combinations of the two
linear functions are used in the objective function of problem (MP2). This modification
makes it easier to find critical parameter values, since the interval [0, 1] over which the
auxiliary parameter ranges is bounded. The rest of the method remained the same.
Although Konno and Kuno (1992) did not explicitly say it, their algorithm can be
viewed as searching the efficient extreme points of problem (BCP) for one that is a global
optimal solution of problem (LMP2). Notice that for a sufficiently small value ', an
extreme point optimal solution (x', 4') to problem (MP2) coincides with an optimal
solution x' of the linear program min {(c',x)+d x D Similarly, for a sufficiently
large value [', an extreme point optimal solution (x', ') coincides with an optimal
solution x" of the linear program min {(c',lx)+ dxe DI. For any fixed value > 0, the
objective function F(x,4) is a composite objective function formed by multiplying the
two linear functions by positive values and summing the result. It is well known that any
extreme point minimizer of such a composite objective function over the feasible region
D is an efficient extreme point of the problem (BCP) (Steuer 1986). The efficient
extreme points of problem (BCP) are found by solving linear programs for parameter
values between and 4'. As Aneja, Aggarwal, and Nair (1984) have shown, the global
solution lies at an efficient extreme point of D in problem (BCP).
A disadvantage of the algorithm of Konno and Kuno is that it may require many
pivots to solve problem (MP2) for all possible parameter values. This will especially be
true if there is a great conflict between the two linear functions of the objective function.
If for example c2 = c,, then every extreme point of D is an efficient extreme point of
problem (BCP). Since the size of the set of extreme points of the polytope D grows
exponentially with D, the number of optimal solutions to problem (MP2) over the entire
range of parameter values grows exponentially with D and is not bounded by a
polynomial. Konno and Kuno in fact observed that the computational time increased as
the number of local minima increased. An additional disadvantage of the Konno and
Kuno algorithm is that many of the pivots performed will be unnecessary when they are
to bases that do not improve on a previously found solution.
In another paper, Konno and Kuno (1990) added a convex function to the
objective function of problem (LMP2) to obtain the problem (CLMP). With this addition,
the objective function may no longer be quasiconcave and therefore, the global minimum
may not necessarily be attained at an extreme point of the feasible region D.
To solve problem (CLMP), Konno and Kuno (1990) proposed an algorithm that
solves a parametric master problem which, for a fixed parameter value, is a nonlinear
convex programming problem. The algorithm involves solving this master problem a
finite number of times, once for each of a finite number of prechosen values for the
parameter. A troublesome aspect of the algorithm is that it is difficult to determine the
proper parameter values to choose. The authors suggested choosing values for the
parameter that are equally spaced in the interval of possible parameter values and solving
the resulting master problems to determine a neighborhood containing a globally optimal
solution to problem (CLMP). A local search is then done in that neighborhood for a
globally optimal solution using the KarushKuhnTucker conditions. Care must be taken
however, to attempt to define the spacing between the points to be small enough so that a
global optimal solution is not missed.
25
The difficulty that Konno and Kuno (1990) encountered in their method in
determining parameter values can be eliminated if we assume that the convex function g
in the objective function of problem (CLMP) is a linear function. Problem (GLMP) is
obtained by making this replacement. Konno, Yajima, and Matsui (1991) considered
problem (GLMP), but they assumed that d, and d, are zero. To solve problem (GLMP),
Konno, Yajima, and Matsui formulated the master problem
(MP3) min F(x;4) = (co,x) + (c2,x),
s.t. xe D,
(c', )= 0.
Notice that the parameter appears in both the objective function and in a righthand
side of a constraint.
Konno, Yajima, and Matsui (1991) showed that x' is a global solution of problem
(GLMP) if (x', E') is an optimal solution of problem (MP3). Schaible and Sodini (1995)
used problem (MP3) to show that a global optimal solution of problem (GLMP) lies on
an edge of D.
Konno, Yajima, and Matsui (1991) proposed a parametric simplex algorithm that
includes a righthand side analysis and an objective function analysis to determine
intervals of parameter values for which bases remain both feasible and optimal. The
parametric analysis sweeps through parameter values from 4, = min{(c',x)Ix e D} to
a, = max(c', x)Ix e D. The objective function F is then minimized over each of the
intervals.
26
Konno, Yajima, and Matsui (1991) tested their algorithm on randomly generated
problems of up to 350 constraints and 300 variables. They found that the problems can be
solved in much the same computational time as that of solving linear programs of equal
size.
The algorithm of Konno, Yajima, and Matsui (1991) suffers from the same
disadvantages as the algorithm of Konno and Kuno (1992). In particular, its efficiency
depends on the number of pivots performed to solve problem (MP3) for all possible
parameter values. Also many of the pivots performed will be unnecessary when they yield
bases that do not improve on a previously found solution.
Schaible and Sodini (1995) improved the algorithm of Konno, Yajima, and
Matsui (1991). From a given simplex tableau for problem (MP3), Schaible and Sodini
used parametric analysis to derive a formula that calculates the value of the objective
function F as the constraint (c', x) = is set to increasing values of i'. As !' increases,
parametric righthandside analysis calculates new values for the basic variables. Schaible
and Sodini then derived some optimality conditions that detect when the parameter 4' is
set to a value such that from an optimal solution (x', ') of problem (MP3), one obtains a
local minimum x' of problem (GLMP). By applying these optimality conditions,
Schaible and Sodini were able to develop a simplexbased algorithm that solves problem
(MP3) in a finite number of primal and/or dual simplex iterations.
The algorithm proposed by Schaible and Sodini (1995) has three advantages over
the algorithm of Konno, Yajima, and Matsui (1991): (1) It may terminate before the
maximum possible parameter value m, has been reached. (2) It is more efficient in that
it may skip over local optimal solutions that do not improve the objective function value.
(3) It can be used even when the feasible region is unbounded, and it can detect when
problem (GLMP) is unbounded from below.
Muu and Tam (1992) also considered problem (CLMP), but in their work, the
feasible region D is relaxed to a compact convex set. They seem to be the only
researchers to have considered this generalization of problem (CLMP). The authors
however tested their algorithm using a polytope for the feasible region.
Muu and Tam (1992) formulated the parametric master problem
(MP3') minF(x;4)= g(x)+4lc2,x)+d2 ,
s.t. x D,
c',x) +d,=(, i0.
They proposed a branch and bound algorithm to solve problem (MP 3'). Branch and
bound is a technique commonly used by algorithms in global optimization. Branching
refers to the successive partitioning of the feasible region and bounding refers to the
computation of lower and upper bounds on the global optimum over the partitions.
Partitions of the feasible region that produce a lower bound on the objective function that
exceeds the best upper bound found so far by the algorithm are eliminated from further
consideration. Such partitions are said to be fathomed. A branch and bound algorithm
terminates when all of the partitions have been fathomed.
In the algorithm of Muu and Tam (1992), partitions of the feasible region are
constructed by restricting the value of (c', x) + d, to values within an interval. The
algorithm begins by finding an interval o := [i,, 2] of achievable values of (c, ) + d
by solving the two convex programs 4, : min (c', x) + d,lx Dj and
S:= max {(c', x) + dix D}. Optimal solutions uo and vo are then obtained for the two
convex programs
(,):= min ,c',x +d2+ g(x)xE D, [c',x)+d,]i } i=1,2.
A lower bound 3(Io) over the interval I, of the objective function F of problem
(MP 3') is found by selecting /3(I,):= min {f3(4), 3(2)} An upper bound a, on F is
obtained by selecting := min f (uO), f(vO)}. The interval I0 is next bisected and the
procedure repeated using the two subintervals. A subinterval that produces a lower bound
that exceeds the current upper bound is eliminated from further consideration; i.e. that
subinterval is considered to be fathomed. The procedure continues bisecting intervals Ik
to generating a sequence of solutions {k =' that converge to a limit point x' that is a
global optimal solution. Computational experiments on problems up to (m, n) = (30, 200)
showed that the algorithm is very efficient when both vectors c and d are positive.
2.2.4. Methods Based on Polyhedral Annexation
A limitation of conventional optimization methods is that they can become
trapped at a local minimum, or even a stationary point, if they are applied to a global
optimization problem, e.g. see the algorithms proposed by Swarp (1966a, 1966b). The
central problem of a global optimization method then is to overcome this limitation by
providing a certification test for global optimality, and if a point is not globally optimal,
determining how to move to a better solution. Tuy (1991) called this the subproblem of
29
"transcending the incumbent" where the incumbent is the best feasible solution found so
far by an algorithm.
Let f be the objective function for problem (LMP2), and let X be a vertex of D
that represents the incumbent solution for this problem. Then, from Tuy (1991), to
transcend the incumbent, one must find a point in x e D such that f(x) < f () or else
establish that no such point exists, i.e. that i is a global optimal solution for problem
(LMP2).
Let G := {x e f (x) f (1)}, where S is a convex set containing D. The
problem of transcending the incumbent can then be restated as the following problem.
(GCP) Check if Dc G and if not, find a point x e D \ G.
Problem (GCP) is known as the Geometric Complementary Problem.
Tuy (1990) developed the method of polyhedral annexation to solve problem
(GCP). In polyhedral annexation a sequence of polytopes P, c P2 c * c Pk c is built
by adding a vertex to the polytope Pk_ of the previous iteration in such a way that a
vertex of D is annexed into the new polytope Pk. The sequence P n D, P2 n D,... forms
an expanding inner approximation of D. When a polytope P.h D is found, all of the
extreme points of D have been searched and the algorithm terminates. Associated with
the sequence of polytopes P, c P, c ... c Pk c ... is the sequence of their polars
P* > P,* D .. D PkW* D .. where a polar E' of a convex set E in R" is defined as
E* := R"(y, x) I for allx e E}. A dual correspondence exists between the facets of
a polytope Pk and the vertices of its polar Pk'. The subproblem of determining the
inequality representation of Pk, after a new vertex has been added can then be solved by
solving the easier problem of computing the vertices of Pk. The termination condition
P, D D has the corresponding condition P*' D'. For a more detailed description of
polyhedral annexation, see the chapters on inner approximation in Horst, Pardalos, and
Thoai (1995) or in Horst and Tuy (1993).
Tuy and Tam (1992) proposed two algorithms that are derived using the
polyhedral annexation method with a dualization and dimension reduction technique
developed by Tuy (1991). Dualization refers to solving the original problem by solving
the dual problem of generating a sequence of polars until a polar P,' g D* is found. The
key to the dimension reduction technique is the introduction of a cone into problem
(GCP). Tuy and Tam (1992) assumed that c' and c2 are linearly independent vectors and
then formed the cone K : x R" (c',x) 0, i =1, 2}. Cone K is of interest since if
xe D is an incumbent solution, then, for any e (I + K), f(i) 2 f(x). In other words,
cone K identifies points in R" that can do no better than the incumbent solution x.
Computational effort might be saved using cone K since a part of the feasible region D
can be eliminated from further consideration and the search narrowed to the remaining
portion of D.
The first algorithm proposed by Tuy and Tam (1992) solves problem (LMP2) by
solving problem (GCP) through the dualization process of generating a sequence of
polars until a polar P, c D' is found. Tuy and Tam (1992) showed that the polar K' of
cone K is explicitly given as K' = {y R"Iy = tc' tc2 for some t 20, t, 2 Oj. Any
vertex I in a polar Pk lies in the polar cone K', and the multipliers i, and f, used to
express 9 are unique, since c' and c2 are linearly independent vectors. Polar cone K' is
used to solve the dual problem by building a collapsing sequence of polars
P,' >) P2* ... 3 P,* =) ** with each polar being an improved approximation of D'. The
search is conducted in the twodimensional space generated by c' and c2 rather than in
the original ndimensional space. Solving the linear program
(LP(f)) max {,(c',x)i2(c2,x )xe D,
where t1 and 2, are the multipliers used to express some vertex I =Tc' t2c' of P,
tests for the termination condition P, c D'.
The second algorithm proposed by Tuy and Tam (1992) is motivated by the
observation that for a fixed value of t = (t, t2), problem (LP(t)) is equivalent to the linear
program
(LP(a)) max {(c'a(c c',x)xe D}
where a = t/(t, + t,)e [0, 1]. The first algorithm thus reduces to solving a sequence of
linear programs (LP(a)) for different values of the parameter a. The second algorithm
proposed by Tuy and Tam (1992) is to parametrically solve problem (LP(a)) for all of
the critical values of a at which new bases become optimal. The objective functionfof
problem (LMP2) is evaluated at each basis and a global optimal solution chosen from
those bases. The second algorithm of Tuy and Tam (1992) is essentially the same
parametric problem (MP2) used by Konno and Kuno (1992).
Tuy and Tam (1992) ran computational experiments using both the first
polyhedral annexation algorithm and the second parametric algorithm. Their results
showed that for solving problem (LMP2), the parametric algorithm performed better than
the polyhedral annexation algorithm. The polyhedral annexation algorithm is not as
efficient because more simplex pivots were required than for the parametric algorithm.
Tuy and Tam (1992) proposed an improved variant of the polyhedral annexation
algorithm that reduces the number of pivots and the number of objective function
evaluations. The authors observed that the improved algorithm may potentially be more
useful for a problem with an objective function that is difficult to evaluate. The
computational experiments run using the parametric algorithm on problems of up to (m,
n) = (30, 200) and positive problem data were in line with the results reported in Konno
and Kuno (1992).
2.3. Extensions of Algorithms for Problem (LMP2) to Solve Problem (LMP)
when p 3
The polyhedral annexation method of Tuy and Tam (1992) and the outcomespace
algorithms of Thoai (1991) and Falk and Palocsay (1994) can be extended to the more
general problem (LMP) where p 3. Although the algorithms remain unchanged, the
subproblem of determining the new vertices becomes more difficult as the number of
function terms in the objective function increases.
2.4. Methods to Solve Problems (CMP), (GCMP), and (CCMP)
Relatively little work has been done in designing exact global solution algorithms
that address problems (CMP), (GCMP), and (CCMP). The algorithms that have been
proposed fall into two categories: (1) methods based on solving a reformulated problem
and (2) a method based on outer approximation.
2.4.1. Methods Based on Solving a Reformulated Problem
Konno and Kuno (1992) introduced problem (CMP) where p = 2 and formulated
a master problem by introducing a parameter into the original problem to separate the two
functions of the objective function into a summation. This technique of embedding the
original problem into a problem in a higher dimensional space is similar to the one used
by the authors in the same paper to solve problem (LMP2). At the time, Konno and Kuno
were not able to give an algorithm for solving the master problem. In Kuno and Konno
(1991) the authors proposed a branch and bound algorithm along with an underestimation
function to solve it. Computational results for problems of up to (m, n) = (200, 180)
indicated that the algorithm is efficient when the objective function is the product of a
linear function and a quadratic function and the feasible region is a polytope.
Kuno, Yajima, and Konno (1993) extended the paramaterization technique of
Kuno and Konno (1991) for problem (CMP) to handle cases where p 2. They showed
that a global optimal solution to problem (CMP) can be obtained by solving the
equivalent problem
(MP4) min minG(x;4)= J4,(x),
where = ( ~ l 0}. For a fixed , let x'() denote an optimal
solution of min G(x;})= F ,f,(x). Let h: E > R be defined by h(4):= G(x'( ); ) for
j=I
34
any E E E. Solving problem (MP4) then reduces to solving the problem in RP given by
(MP4') min h().
Kuno, Yajima, and Konno (1993) showed that h is a concave function over 5 and
therefore a global optimal solution of problem (MP4') exists on the boundary of E They
proposed an outer approximation method for solving problem (MP4') and tested their
algorithm against two subclasses of problem (CMP): (1) problem (LMP) and (2)
problems similar those tested in Kuno and Konno (1991) in which the objective function
is the product of a linear and a quadratic function and the constraints are linear
inequalities. Computational experiments showed that the total computational time is
dominated by that needed for solving the convex minimization master problems for each
parameter value. The results also showed that the number of cuts and vertices generated
increases rapidly as p increased from 2 to 5. The authors asserted that this was due to
inefficiencies in computing new vertices, especially when p exceeds 5. However, if p is
held constant, these numbers increased very slowly as the number of constraints and
variables increased. The authors concluded that their algorithm is reasonably efficient
when p is less than 4.
Jaumard, Meyer, and Tuy (1997) added a convex function to the objective
function of problem (CMP) to form problem (CCMP). The authors showed that problem
(CCMP) can be reduced to a quasiconcave minimization problem in RP that is a
generalization of problem (MP4') used by Kuno, Yajima, and Konno (1993). In the
special case where fo = 0 in problem (CCMP), the reduced quasiconcave minimization
problem in Jaumard, Meyer, and Tuy (1997) can be shown to be equivalent to the one
35
used by Kuno, Yajima, and Konno (1993). Jaumard, Meyer, and Tuy (1997) find a global
solution of problem (CCMP) by finding an optimal solution to the quasiconcave
minimization problem in RP using a conical branch and bound method. They ran
computational experiments using their algorithm on test problems similar to those used
by Kuno, Yajima, and Konno (1993) and Thoai (1991). The authors report that their
results are very sensitive to the magnitude of p and not as sensitive to the size (m, n) of
the constraint matrix.
Sniedovich and Findlay (1995) analyzed problem (CMP) from the perspective of
cprogramming but did not give a complete algorithm for solving it. Cprogramming is a
technique developed by Sniedovich (1984) for solving an optimization problem of the
form
(CP) q:=mixn ((p(x)),
where X is some nonempty set, qp is a mapping on X with values in RP, and y is a
differentiable and pseudoconcave function on some open set containing the set
Tp(X): = {q(x) x e X}. The heart of the technique is to linearize the function V and
transform the original optimization problem into the parametric programming problem
(MP5) q(4):= minei(x)4, E RP.
Sniedovich showed that if x' is a globally optimal solution for problem (CP), then an
optimal solution '* for problem (MP5) is = Vy(q,(x')), where Vy(.) is the gradient
of f.
36
For problem (CMP), the objective function can be expressed as the composite
y(qp) of two functions, where, for each xe R", tp(x)= (f,(x), f,(x )... f,(x)), and, for
each y R y(y)= l y,. Sniedovich and Findlay claimed without proof that Vy is a
1=1
differentiable and pseudoconcave function on the open convex set 4 e R > 0}. Since
problem (CMP) satisfies the requirements of cprogramming, it can be solved by solving
the parametric problem
(MP5') q( ):=min i f;i (x eE,
.X i=l
where E is any subset of RP such that Vy(q,(x))e E for all xe X. In problem (MP5'),
the parameter appears only in the objective function, whereas in problem (MP4) the
parameter I appears in both the objective function and in the constraints. Standard
Lagrangian methods can be employed to solve problem (MP5') for all e EZ, while
specialized methods are required to optimize the objective function of problem (MP4)
with respect to the original variable x and the parameter 4.
Kuno and Konno (1991) and Konno, Kuno, and Yajima (1994) considered
problem (GCMP) for cases where q = 1 and q > 1 respectively. For q = 1, the master
problem and solution algorithm are similar to the one used by Kuno and Konno (1991) to
solve problem (CMP) when p = 2. Computational experiments showed that the
underestimation function does not perform as well as it does for problem (CMP).
For q 2 1, the master problem in Konno, Kuno, and Yajima (1994) is formulated
by introducing a pair of parameters for each pair of convex functions that appear in the
37
objective function of problem (GCMP). The master problem is a convex minimization
problem in the space R"2' and is solved using an outer approximation algorithm.
Computational experiments conducted using a polyhedron for the feasible region showed
that for q = 1, this algorithm required less than half the computational time required by
the branch and bound with underestimation function algorithm proposed in Konno and
Kuno (1992) to solve problem (CMP).
Tuy (1992) gave problem (CMP) as an example of an optimization problem that
can be formulated as a Geometric Complementary Problem and solved it using a
parametric programming problem. The parametric programming problem is a convex
minimization problem in which a positive parameter vector is used to build a composite
objective function from the convex functions in the objective function of problem (CMP).
A complete algorithm that includes solving the parametric program was not given.
2.4.2. A Method Based on Outer Approximation
Thoai (1991) extended the algorithm based on the outer approximation technique
that he proposed for solving problem (LMP2) to address the solution of problem (CMP)
when p= 2. The main idea is to build a sequence of decreasing polytopes
P0 D Pi 3 DX of the convex feasible region X and a sequence of decreasing
polytopes So D S 3 ) Y of the outcome set Y, where
f = {ye R2ly, = f,(x) y2 = f2(x)for some x X.
Problem (CMP) is then solved by applying a modified version of the algorithm for
problem (LMP2). In any iteration k, up to two cuts are introduced, one for Pk and one
for Sk, to obtain tighter approximating sets.
Since the algorithm does not depend on the actual value of p, it can be extended
to handle cases where pt 3.
2.5. Methods to Solve Problem (LMP) as a Concave Minimization Problem
Konno and Kuno (1992) showed that the objective function of problem (LMP) is
not a convex function over the feasible set D. Therefore, problem (LMP) is not a convex
programming problem. However, since the natural logarithm function In is a strictly
increasing concave function on (0, o), it is easy to show that the function
F(x)=ln[ (c',x)+d, + = ln Ic',x)+d,]
defined for all x D is a concave function. In addition, the optimal solution set of the
concave minimization problem
(CMIN) min F(x s.t.xe D,
is identical to the optimal solution set of problem (LMP). Therefore, any concave
minimization method may be applied to problem (LMP) if the objective function is
replaced by its logarithmic equivalent.
Using the above transformation modification, Tuy (1991) showed that problem
(LMP) could be solved in a reduced dimension space using polyhedral annexation and the
dualization and dimension reduction technique. The algorithm presented in Tuy and Tam
(1992) is essentially an improvement of the one in Tuy (1991).
Ryoo and Sahinidis (1996) also converted problem (LMP) into the problem
(CMIN). To solve problem (CMIN), they employed a branch and bound algorithm that
incorporates the use of valid inequalities to accelerate convergence. Branch and bound
algorithms may slowly converge to an optimal solution when the gap between the initial
upper and lower bounds is large. A valid inequality is a inequality constraint that does not
exclude any solution that yields an objective function value lower than the current best
upper bound. By introducing valid inequalities into the constraint set, inferior parts of the
feasible region may be removed from further consideration without eliminating possible
global optimal solutions. A second use of valid inequalities is to reduce the range of
values that the variables in the problem can assume. Ryoo and Sahinidis referred to these
two uses of valid inequalities as range reduction mechanisms. The performance of the
bounding procedure in the branch and bound algorithm is improved by using these range
reduction mechanisms, since smallersized partitions of the feasible region are used and
the variables are restricted to reduced ranges of values.
Ryoo and Sahinidis implemented the branch and bound algorithm along with the
range reduction mechanisms in a computer program called BARON (BranchAndReduce
Optimization Navigator). To more easily calculate lower bounds on the objective function
F of problem (CM IN i ,\ er a partition of the feasible region, the authors replaced F by a
linear underestimating function. Lower bounds were then calculated by solving linear
programs. The authors tested randomlygenerated problems in sizes from (m, n) = (50,
50) to (200, 200), with p ranging from 2 to 5. They reported that only a small fraction of
the total CPU time is consumed in the range reduction mechanisms and that there seemed
to be a loworder polynomial relationship between the CPU time and the value of p.
CHAPTER 3
CONCAVE MULTIPLICATIVE PROGRAMMING PROBLEMS: ANALYSIS AND
AN EFFICIENT POINT SEARCH HEURISTIC FOR THE LINEAR CASE
3.1. Introduction
An important, but little researched area that deserves more attention, is the
development of heuristic algorithms for finding a good solution for multiplicative
programming problems. In some applications, a good, though not necessarily globally
optimal solution, may adequately meet the requirements of a user (Konno and Inori
1989). In these cases, since multiplicative programming problems are known to be NP
hard, the expenditure of computational effort required to globally solve them may not be
needed.
This chapter has two purposes. The first is to present an analysis of problem (Px)
when problem (Px) is a concave multiplicative programming problem. The second
purpose is to propose a heuristic algorithm designed for the case where problem (Px) is a
linear multiplicative programming problem.
The analysis of the concave multiplicative programming problem is presented in
Section 3.2. This analysis shows a new way to write a concave multiplicative
programming problem as a concave minimization problem and some theoretical
consequences of this. It also shows some relationships between concave multiplicative
programs and certain multipleobjective mathematical programs. In Section 3.3, by using
some of the results of Section 3.2, we present and explain the workings of an efficient
40
41
point search heuristic algorithm that we have developed for the linear multiplicative
programming problem. Section 3.4 reports and analyzes some statistics summarizing the
computational results that we obtained by coding the heuristic algorithm and applying it
to 260 randomlygenerated linear multiplicative programs. In Section 3.4 we also report
the results of applying the heuristic algorithm to a multiplicative programming problem
formed from a decision situation using real data. In Section 3.5, we discuss the major
results of this chapter.
3.2. Analysis
Assume in problem (P,) that X is a convex set and that, for each j = 1, 2,..., p,
f : X 4 R is a concave function; i.e., assume that problem (Px) is a concave
multiplicative programming problem. Consider the function : X  R defined for each
xe X by
g(x)= log g(x).
Then, it is a simple matter to show that k : X  R is a concave function and that the
optimal solution set to the concave minimization problem
min k(x), s.t. x X, (3.1)
is identical to the optimal solution set of problem (Px). Thus, any concave multiplicative
programming problem of the form of problem (Px), if rewritten in the form (3.1), can be
solved by applying any appropriate generalpurpose concave minimization algorithm to
(3.1). For discussions and reviews of concave minimization algorithms, see, for instance,
Benson (1995), Benson (1996), Horst and Tuy (1993), and Pardalos and Rosen (1987).
It is interesting and useful in both practice and theory to observe that, in addition
to (3.1), there is at least one other way to rewrite a concave multiplicative programming
problem as a concave minimization problem. To show how this can be accomplished, we
will first prove the following preliminary result.
Lemma 3.2.1. Let a RP satisfy a > 0, and consider the nonlinear programming problem
v = min(a, ), s.t. Ae A, (3.2)
where A = e Rf AR j 1, Al 0 Then, v is finite and problem (3.2) has at least one
optimal solution.
Proof. Notice that, if Ae A, then A > 0 and (a,A) > 0. Therefore, v > 0. This,
combined with the fact that A 0, implies that v is finite.
Now, suppose that, for each j = 1, 2,..., there exists a vector Aj e A such that
(a,Aj) v+ej,
where {e, t is a strictly decreasing sequence of positive real numbers such that
lime = 0. Then the sequence {, is either bounded or unbounded.
Case 1: {A [, is bounded. Then, for some bounded set XA A, X' E A for each
j = 1, 2,.... Therefore, by passing to an appropriate subsequence {M ,.' of {(I ,, if
necessary, we can guarantee that X = limAj exists. Furthermore, since V E A c A for
each j J, and A is a closed set, Aelongs to By assumption,
each j e J, and A is a closed set, A belongs to A. By assumption,
43
for each j E J. By taking the limits over je J on both sides of (3.3), we conclude that
(a, ) 5 v. Since 1~ A, this implies that X is an optimal solution to (3.2).
Case 2: {I 1}, is unbounded. Then, for some subsequence {,2'}je of {ij,, and
for some k E {l, 2,...p, lim). =+o. For each je J, since X E A, j > 0.
Combined with the fact that a > 0, implies that, for each j e J,
0
By assumption, for each je J,
(a,. ') v+e,. (3.5)
From (3.4) and (3.5), we obtain
a,,
for each je J. By taking the limits over je J on both sides of (3.6), we conclude that
+ o = v, which is a contradiction. Therefore, this case cannot hold, and the proof is
complete. C
Using Lemma 3.2.1, we may now establish the following theorem.
Theorem 3.2.1. Assume in problem (Px) that X is a convex set and that f : X > R,
j = 1, 2,..., p, are concave functions. Let g : X > R be defined for each x eX by
W(x)= pl ifj(x) .
[I' J
Then : X > R is a concave function.
44
Proof. Consider the function h: X + R defined for each xe X by
h(x)= min Ajf,(x), s.t.Ae A, (3.7)
j=1
where A is as defined in Lemma 3.2.1. From Lemma 3.2.1, since f, is strictly positive
on X for each j = 1, 2,..., p, it follows that the minimum in (3.7) exists and is finite for
each xe X. If, for each A e A, we define a function hA :X 4 R by
h, (x)= f,(x),
J=I
then for each xe X, h(x) may also be written as
h(x)= min h (x). (3.8)
Notice that, for each A e A, hA : X 4 R is a concave function. From this and (3.8), we
conclude that h: X  R is also a concave function (Rockafellar 1970).
To complete the proof, we will show that, for each xr X, h(x)= g(x). Toward
this end, fix xe X, and let A(x)e X denote an optimal solution to problem (3.7). From
the KarushKuhnTucker necessary conditions for this problem (Bazaraa, Sherali, and
Shetty 1993), since A(x)> 0, it follows that there exists a nonnegative constant O(x)
such that
fJ(x)9(x) A fk (x)]/,(x)=0, j=1,2,..., p. (3.9)
Since ,A(x)E X is an optimal solution to problem (3.7), it is easy to see that
kI (x)=1.
k~l
Together with (3.9), this implies that
,Ai(x)f1(x)= (x), j= 2,.... p. (3.10)
From (3.10), it follows that
A,(x)=O(x)/fj(x), j=1,2,...,p.
By substitution in
I,(x)= 1,
this implies that
e(x)== f (x) (3.11)
From equations (3.10) and (3.11), we see that
Zx)f,(x)= p f(x) (3.12)
Since xe X and A(x)e X is an optimal solution to (3.7), the lefthandside of equation
(3.12) coincides with h(x). By definition of g, the righthandside of equation (3.12)
equals g(x) so that the proof is complete. [
Theorem 3.2.1 can also be proven by using a composite function approach and
showing several preliminary results (Avriel, Diewert, Schaible, and Zang 1987). We offer
the proof here, because it is more direct and because we will use it below to help derive a
corollary of interest.
46
Notice from Theorem 3.2.1 that, when problem (Px) is a concave multiplicative
program, the optimal solution set of problem (Px) is identical to the optimal solution set
of the concave minimization problem
min g(x), s.t.xe X, (3.13)
where g: X  R is defined for each xe X by
(x)= p[g(x) i.
In practice, this implies that any concave multiplicative program (Px), if rewritten in the
form (3.13), can be solved by applying any suitable concave minimization algorithm to
(3.13). Notice also that problem (3.13) is a simpler reformulation of problem (Px) for the
concave case than the typical reformulation used in the literature to solve problem (Px)
in the convex case (see e.g., Konno and Kuno 1992, Kuno and Konno 1991, Thoai 1991,
and Kuno, Yajima, and Konno 1993).
Theorem 3.2.1 also has some interesting theoretical implications concerning the
product of functions. For instance, for any finite set of concave functions fj, j = 1, 2,
..., p, each defined on a common nonempty convex domain X c R" and each strictly
positive on this domain, it is known that the function g: X  R defined by their product
is not necessarily concave, convex, or quasiconvex on X (Kuno, Yajima and Konno
1993 and Avriel, Diewert, Schaible and Zang 1988). However, from Theorem 3.2.1, the
function f :X R given by
for each xe X is a concave function on X.
47
In addition, Theorem 3.2.1 implies the following result concerning the product of
a set of concave functions.
Corollary 3.2.1. Let X and fj, j= 1, 2,..., p, be defined as in Theorem 3.2.1, and
suppose that g: X  R is defined for each xe X by
g(x)= I,(x).
Then g: X  R is a quasiconcave function.
Proof. Choose a R, and let
L,= {xe Xlg(x) a}.
If a 0, L, = X is a convex set. If a > 0, then from Theorem 3.2.1 and Rockafellar
(1970), the set
= I{xe xEp[g(x)'IIP p}
is a convex set, where = pa'l/. Since L = L,, this implies that L, is a convex set.
Therefore, we have shown that, for any a R, L, is a convex set. This is equivalent to
showing that g: X + R is a quasiconcave function (Bazaraa, Sherali, and Shetty 1993),
so that the proof is complete. O3
It follows from Corollary 3.2.1 that any concave multiplicative programming
problem (Px) is a problem involving the minimization of a quasiconcave function over a
convex set. Many of the most popular algorithms for minimizing a concave function over
a convex set are equally suitable for minimizing quasiconcave functions over convex sets
(Horst and Tuy 1993 and Benson 1995). As a result, we see that any concave
multiplicative program (P ) can be solved by applying any number of suitable concave
minimization algorithms directly to problem (Px). In particular, no reformulations of
problem (Px) are needed to apply these algorithms.
Remark 3.2.1. Corollary 3.2.1 has been previously shown to hold for the special case
where p = 2, X is a nonempty, compact polyhedron, and f, and f2 are linear functions
(see, e.g., Konno and Kuno 1992).
The next corollary of Theorem 3.2.1 concerns the minimization problem (3.7)
used in the proof of the theorem. Possible uses for this corollary may include the
construction of methods for finding local optimal solutions to concave multiplicative
programs, although we will not investigate this here.
Corollary 3.2.2. Let X and f, j = 1, 2,..., p, be defined as in Theorem 3.2.1, and let A
be defined as in Lemma 3.2.1. Then, A is a convex set and, for each xe X, the unique
optimal solution A(x) to problem (3.7) is given by
A (x)= f(x) f(x), k = 1,2,..., p.
Proof. Notice that A may be rewritten according to the relation
A= AEintR p A[ I/P (3.14)
where
intR,' ={le RPA>0}.
It is easy to see that, for each j = 1, 2,..., p, h, :int RP  R, defined for each a int RP
by
49
h,(4) = ,
is a concave function on int RP that satisfies
hi(A)> 0, for all AE int Rf.
Therefore, by Theorem 3.2.1, the function m:int R+ R defined for each A e int Rf by
m(A)= p[ ,
is a concave function. This implies that
{Ae intR IRn(A.l)g p
is a convex set (Rockafellar 1970). By (3.14), this proves that A is a convex set.
Now, fix xe X, and let A(x)e A denote an optimal solution to problem (3.7).
From the proof of Theorem 3.2.1, this implies that, for each k = 1, 2,..., p,
,(x)=e(x)/ f,(x),
where O(x) is given by (3.11), so that the corollary is proven. E
In addition to its relationships to concave minimization, a concave multiplicative
program also has some interesting ties to multipleobjective mathematical programming.
In the remainder of this section, we will show some of the theoretical relationships
between concave multiplicative programs and certain multipleobjective mathematical
programs. In the next section, some practical benefits of those relationships will be
demonstrated.
Let f(x) denote the vector
[f,(x), f,.(x)...,
50
where f : X R, j = 1, 2,..., p, are the functions used in defining problem (Px).
Then, the components of the vector f(x) are generally conflicting, in the sense that the
infima over X of fj(x) j = 1, 2,..., p, are generally not simultaneously achieved at the
same point in X. As a result, inherent tradeoffs in the achievable values of the
components of f(x) over x e X are present. To account for these tradeoffs, and to seek
what decision makers call a most preferred solution in situations where the goal is to
attempt to simultaneously minimize f,(x), j= 1, 2,..., p, over X, one of the most
popular approaches is to consider the associated multipleobjective mathematical
program
VMIN f(x), s.t.xe X. (3.15)
In particular, in typical situations, a most preferred solution in X will exist that is also an
efficient solution for (3.15), where an efficient solution is defined as follows.
Definition 3.2.1. A point x E R" is called an efficient solution for (3.15) when xo E X
and, whenever f(x)< f(xo) for some xe X, then f(x)= f(xo).
An efficient solution is also called a nondominated or Paretooptimal solution. By
generating or searching the set XE of the efficient solutions for (3.15), decision makers
are able to observe the inherent tradeoffs among the objective functions fj, j = 1, 2,...,
p, that are available over X and are often able to choose from X. a most preferred
solution. For further discussions on multipleobjective mathematical programming and its
applications, the reader may consult, for instance, Cohon (1978), Evans (1984), Luc
51
(1989), Sawaragi, Nakayama, and Tanino (1985), Stadler (1979), Steuer (1986), Yu
(1985), Zeleny (1982) and references therein.
The first relationship between multiplicative programming and multipleobjective
mathematical programming is given in the following result. The proof of this result is an
elementary exercise.
Proposition 3.2.1. Any optimal solution to problem (Px) must belong to the efficient set
XE of the multipleobjective mathematical programming problem (3.15).
Notice that Proposition 3.2.1 holds for arbitrary multiplicative programming
problems (Px). The next result, however, is restricted to certain types of concave
multiplicative programs.
Proposition 3.2.2. Assume in problem (Px) that X is a compact, convex set and that
f, :X 4 R, j = 1, 2..., p, are concave functions. Then, there exists an optimal solution
to problem (Px) which is an extreme point of X.
Proof. From Theorem 3.2.1, problem (Px) can be solved by finding an optimal
solution to the concave minimization problem (3.13), where g': X  R is the concave
function defined by
g(x) = P[if(x) ,
for each xe X. Since X is a nonempty compact, convex set, from Horst and Tuy (1993),
problem (3.13) has an optimal solution that is an extreme point of X. These two
observations together prove the desired result. O
52
Taken together, Proposition 3.2.1 and 3.2.2 imply that any concave multiplicative
programming problem with a compact feasible region has at least one optimal solution
that is an efficient extreme point solution to the multipleobjective mathematical
programming problem (3.15). Special cases of this observations have been alluded to in
the literature (see, e.g., Aneja, Aggarwal and Nair 1984 and Sniedovich and Findlay,
1995). In the next section, we put this observation to practical use.
3.3. Efficient Point Search Heuristic
Assume in this section that, in problem (Px),
X = xe R'"Ax bl
is a compact polyhedron, where A is an m x n matrix and b e R', and that for each
j = 1,2,..., p, f (x)= (c, x), where c' e R" for each j= 1,2,...,p. Then problem
(Px) is a linear multiplicative programming problem or, more briefly, a linear
multiplicative program (Konno and Kuno 1992). We have designed and tested a heuristic
algorithm for this problem, based in part on some of the results in the previous section. In
this section, we will formally state this heuristic algorithm and explain its workings.
The multipleobjective program (3.15) associated with a linear multiplicative
problem may be written as
VMIN Cx, s.t. Ax b, (3.16)
where C is the pxn matrix whose jth row equals (c ), j = 1, 2,...,p. Problem (3.16)
is a multipleobjective linear programming problem (Steuer 1986 and Yu 1985). Let X,
denote the set of extreme points of
53
X =Ixe R'Ax b).
Then, by Proposition 3.2.1 and 3.2.2, an optimal solution to the linear multiplicative
programming problem can be found in the set
x. = (XEnx,)
of efficient extreme points of problem (3.16). The set X,x is finite, and various
procedures have been developed for generating it in its entirety (see, e.g., Steuer 1986, Yu
1985 and Steuer 1983).
It follows that, in theory, at least, a global optimal solution to a linear
multiplicative problem can be found by completely enumerating the set XE, of efficient
extreme points of the associated multipleobjective linear programming problem (3.16)
and, from this set, choosing the points) with the smallest value of
g(x)= (c'J,x
j=I
(see, e.g., Sniedovich and Findlay 1995). Unfortunately, as we shall see later, in practice
the exponential growth in the size of XE, as a function of problem size (Steuer 1986)
renders this approach impractical for many cases.
The approach of the heuristic algorithm is to efficiently search a dispersed,
carefully chosen sample of candidate points from X Ex in order to find an attractive
solution to the linear multiplicative programming problem. To describe and explain the
workings of the heuristic, we must first present some theoretical background from the
theory of multipleobjective linear programming.
Let
W ={we RI'(e,w)
where ee R' is a vector with each entry equal to 1.0, and M is a positive real number.
For sufficiently large M, from Philip (1972) it is known that a point xO belongs to the
efficient set XE of (3.16) if and only if xo is an optimal solution to the weightedsum
problem
min (wC, ), s.t. Ax < b, (3.17)
for some w = w e W. We will assume henceforth that M is chosen to be large enough to
guarantee that this property holds. It is also well known that the efficient set XE for
(3.16) is given by
XE = U{Xjwe w},
where, for each we W, X, denotes the optimal solution set of the linear program (3.17)
(Steuer 1986 and Yu 1985). Since the optimal solution set to (3.17) for any we W is a
face of
X= E R"'Ax5b\,
it follows that the efficient set XE for (3.16) is equal to the union of the faces X,,
we W, of X. Although XE is a connected set (Yu 1985), it is generally nonconvex. The
heuristic algorithm will individually identify efficient faces X,, we W, of X, and find
an approximatelyoptimal extreme point solution to the problem
min (c',x, s.t. x X, (3.18)
j=l
for each efficient face X, that it finds.
Let
Y={ye R'ly=Cx,forsomexe X},
Y = {ye Rly > y, forsomeF Y}.
To aid in its search, the heuristic algorithm will solve the linear program
min w'C,x) (3.19a)
s.t. Cx y, (3.19b)
Ax b, (3.19c)
for various values of y e Y' and we W The heuristic relies in part upon the properties
of problem (3.19) given in the next three results. The first two results follow easily from
Benson (1978).
Theorem 3.3.1. Suppose that xo e R" and let yO = Cxo. Then, xO is an efficient solution
for (3.16) if and only if, with y = yO, xo is an optimal solution to (3.19) for every
we W.
Theorem 3.3.2. If ye Y" and we W, then (3.19) has at least one optimal solution, and
any optimal solution for (3.19) is an efficient solution for (3.16).
Theorem 3.3.3. Suppose in (3.19) that w = w W and that y = yO = Cxo, where xo is
an efficient solution for (3.16). Let (uor, zor) denote any optimal solution to linear
programming dual of (3.19), where uo represents the dual variables corresponding to the
constraints Cx yO of (3.19). Let wO = u + w and let vo = (w CxO. Then, xo belongs
to the efficient face X. of X, and X, can be represented as
56
X, ={xe X(woCx=vo}.
Proof. To prove the theorem, we will show that, with w = 0o, xo is an optimal
solution to problem (3.17). Suppose in (3.19) that w= w e W and that y = yO = Cxo,
where xO is an efficient solution for (3.16) given in the theorem. The dual linear program
to (3.19) is then given by
max(yO,u)(b,z),
s.t. CTuArZ=CTWo
u,z O.
From Theorem 3.3.1, xo is an optimal solution to (3.19) when w = w and y = yO. By
the duality theory of linear programming (Murty 1983), since (UOT, zor) is an optimal
solution to the linear programming dual of (3.19) when w = w and y = yO, this implies
that
(we Cxo = (y, u)(b, z.
By rearranging this equation and using the definitions of yO and WO, we obtain
( CxO C =(b, z). (3.20)
With w = Vo, the dual linear program to (3.17) may be written as
max(b,z), (3.21a)
s.t. ATz = CrWO, (3.21b)
z 0. (3.21c)
Let z denote an arbitrary feasible solution to problem (3.21). From the definitions of u
and w0, this implies that (uor, z) is a feasible solution to the dual linear program of
(3.19). Since (UOT, ZT ) is an optimal solution to the latter problem, it follows that
(y, uO)(b, zo) (yo, )(b, ),
or, equivalently,
Notice that, since (uor,zOT) is an optimal solution to the dual linear program to (3.19), z
is a feasible solution to (3.21). By the choice of z, the preceding two statements imply
that z is an optimal solution to (3.21). Since xo is an efficient solution for (3.16), with
w = W, xo is a feasible solution for (3.17). From (3.20) and the duality theory of linear
programming (Murty 1983), since z0 is an optimal solution to (3.21), this implies that,
with w = wt, xo is an optimal solution to (3.17), and the proof is complete. O
Notice in Theorem 3.3.3 that, for any t > 0, X, = X,. This implies that, in
Theorem 3.3.3, when W" o W, there exists a tE (0, 1) such that tfw e W and
X, = X,,. Thus, in Theorem 3.3.3, when wO e W, X0, has an alternate representation
X, for which w= e W. For simplicity, we may and will assume without loss of
generality that in Theorem 3.3.3, wo e W.
To generate various points ye Y' for use in problem (3.19), the heuristic
algorithm will rely upon the two concepts defined in the next two definitions (see, e.g.,
Zeleny 1982).
58
Definition 3.3.1. The point y' RP is called the ideal point of Y when, for each
j = 1, 2,..., p, y' equals the minimum value of y, over Y.
Definition 3.3.2. The point y" e RP is called the antiideal point of Y when, for each
j= 1, 2,... p, y' equals the maximum value of y, over Y.
Notice that y' and yj' generally do not belong to Y. The algorithm uses these
two points as anchor points in an initialization procedure whose goal is, in part, to
generate a dispersed sample of points from Y'.
The heuristic algorithm may be stated as follows.
Algorithm 3.3.1. Efficient Point Search Heuristic Algorithm
Initialization Phase. See Steps 1 through 5 below.
Step 1. Find the ideal and antiideal points y' and y" of Y.
Step 2. Find an optimal solution [(xj ,Ja' R"' to the linear program
max a,
s.t. y" +a(y' yAI) Cx,
Ax b,
a>O,
and set y* = yA +a'(y' yA).
Step 3. Choose a positive integer S and, for each i = 1, 2,..., S, let
y' =y"+(i/SXy* y ).
Ste 4. Choose a positive integer N such that 1 N < M p+1, let w = e R', and,
for each j = 1, 2,..., p, define w' e R" by
59
I fl, ifi j,
N, if i = j.
Ste 5. Set UB = +o, i =0 and j =0.
Efficient Point Search Phase. See Steps 1 through 6 below.
Step 1. Set y = y' and w= w', and find any optimal solution x" to linear program
(3.19).
Step 2. Set y = Cx and w = w in (3.19), and compute any optimal solution [(u;,
(z' J to the dual linear program to (3.19), where u" denotes the optimal dual variables
corresponding to the constraints Cx < y of (3.19).
Step 3. Let 'j = u'j + w. If w~ is a positive multiple of W'i for some i' i and j' j
such that (i', j') (i, j), then go to Step 6. Otherwise, continue.
Step 4. Let v, = (W!J C"'. For each h = 1, 2,...,n, calculate a, according to the formula
a, = it=l[ c',.V d ]c, (3.22)
and find any basic optimal solution xd to the linear program
min(a, x), (3.23a)
s.t. (wr TCx =v,, (3.23b)
Ax < b. (3.23c)
Ste 5. If fI (ck x )>UB, go to Step 6. Otherwise, set ,= x", and UB= (c*,.
k=1 k*=
and go to Step 6.
60
Step 6. Set j = j +1. If j p, go to Step 1. Otherwise, set i = i +1 and j = 0. If i S,
go to Step 1. Otherwise, Stop: xe XE,, is the recommended solution to the linear
multiplicative programming problem.
In the initialization phase of the algorithm, samples of points from Y' and from
Ware generated. To generate the sample of points from Y', Step 2 of this phase
determines the point y' between y"A and y' such that, of all line segments with
endpoints y" and y that lie in Y' and for which y lies on the line segment connecting
y" and y', the line segment L connecting y" and y' has maximum norm. The sample
y'i = 1, 2,.... S of points from Y" is then generated in Step 3 of this phase by
partitioning L into S line segments of equal length, where S is a positive integer chosen
by the user. In Step 4, a sample of p +1 allinteger vectors from W is generated, where
for p of these vectors, the value N of one of the components is chosen by the user from
the set {l,2,...,M p+l}.
Each iteration of the efficient point search phase of the heuristic executes two key
operations. First, it identifies an efficient face X, of X. Second, unless this face has
been previously identified during an earlier execution of this phase, with w = w' in
problem (3.18), by using a firstorder linear approximation to the objective function of
this problem, it finds an extreme point x' of X in this efficient face that is an
approximate optimal solution to (3.18).
Steps I through 3 of the efficient point search phase of the algorithm identify an
efficient face of X. In Step 1, with y = y' Y' and w= w' e W, the linear program
61
(3.19) is solved for any optimal solution PX. By Theorem 3.3.2, this optimal solution
must exist and is an efficient solution for (3.16). In Steps 2 and 3, with y = CxT and
w = wJ in (3.19), the dual linear program to (3.19) is solved to yield the vector uU RP,
and the weighting vector W' = u + w' is computed. From Theorem 3.3.3, the face X,
corresponding to this weighting vector is an efficient face for (3.16) and contains x.
Furthermore, from the same theorem, this face can be written as
X, = {E R" Ax b,(W Cx =v,,, (3.24)
where v, = (W'i CI'. Step 3 checks whether or not X, has been identified during a
previous execution of this phase of the algorithm. If so, the algorithm proceeds to Step 6
to prepare for another possible iteration of the efficient point search phase of the heuristic.
Otherwise, control shifts to Steps 45.
In Steps 45 of the efficient point search phase, problem (3.18) is approximately
solved using a new efficient face X, as the feasible region. In particular, in Step 4, (3.22)
is first used to construct the nonconstant portion of a firstorder Taylor series linear
approximation (a,x) of the objective function of problem (3.18) at x = x e X,,. Next,
using the representation (3.24) of the efficient face X,, an extreme point minimizer xi
of (a, x) over X,, is found by solving the linear program (3.23). Notice that x4 e XE,
(see Rockafellar 1970). In Step 5, the value achieved by x' in the objective function of
the linear multiplicative problem is compared to the smallest value UB found thus far for
this objective function by the search. If x" achieves a smaller objective function value
62
than UB, x' becomes the new incumbent solution x and UB is reduced in value
accordingly.
Notice that the performance of the heuristic algorithm depends in part upon the
number, locations, and dimensions of the efficient faces (3.24) that are searched via
problem (3.23). This, in turn, is partially dependent upon the sizes of the parameters S
and N chosen by the user. The goal is to search as many points of X,., as possible by
generating a variety of distinct efficient faces (3.24) of large dimensions that are
dispersed widely throughout X,. Notice that, since each efficient face identified by the
heuristic is given in the form (3.24) and searched by solving linear program (3.23), the
individual points in XE,. that are searched by the algorithm are searched implicitly rather
than explicitly, i.e., they do not need to be explicitly enumerated.
3.4. Computational Results
The heuristic algorithm described in Section 3.3 has the following attractive
characteristics:
(a) it can be implemented using only linear programming methods;
(b) it generally implicitly searches many efficient extreme points of (3.16) at once
by optimizing over entire efficient faces of (3.16), rather than by explicitly examining
individual efficient extreme points of (3.16);
(c) it allows the user to manipulate the nature and extent of the efficient face
search through the choices for the input parameters S and N;
(d) it finds efficient faces of (3.16) by attempting to globally sample from a
variety of regions of the efficient set.
63
To evaluate the effectiveness in practice of the heuristic algorithm and its features,
we have written a VS FORTRAN computer code for the algorithm and used it to solve
260 linear multiplicative programming problems of various sizes. To execute the code on
these 260 problems, we used an IBM ES/9000 model 831 mainframe computer. As a
further illustration of the effectiveness in practice of the heuristic algorithm, we solved a
multipleobject linear programming problem in forest management that was derived from
a real decision situation using real data.
To implement Step 3 of the initialization phase of the algorithm, we chose to set
S = 4, so that a sample of five points lying between y" and y' in Y" is always
generated in this step. We used a value of N = 9 in Step 4 of the initialization phase to
help generate the sample of p +1 points from W.
To solve the linear programming problems called for by the heuristic, the
computer code uses the simplex method procedures given in the subroutines of the
Optimization Subroutine Library (International Business Machines 1990). These
subroutines employ anticycling rules to handle degeneracy as needed. Therefore, they are
especially appropriate for solving instances of problem (3.23), since these problems
always contain degenerate extreme points.
Let
intR" ={xe R"lx>0j,
and suppose that k is a positive integer. To generate the 260 test problems, we used the
following random procedure. First, for each j = 1, 2,..., p, we generated the elements of
the vector c E R" by randomly drawing elements from the set (1, 2,..., 10). Next, we
64
generated a nonempty, compact polyhedral feasible region X C int R,. This region can
be written as
X= { R'\Px q,l xj < j=, 2,....n,
where P is a k x n matrix, q E Rk, and 4 E R. To accomplish this, first the elements of P
were generated by randomly choosing elements from the set {l, 2,..., 10}. Next, for each
i= 1,2,..., k, the formula
qi P"
was used to calculate q,, and, finally, 4 was chosen according to the rule
4= max {q,i= l,2,...,k}.
Each test problem was constructed to belong to one of four categories, where a
category is defined by the number p of linear functions used in the objective function
S(ci, x) of the test problem. The values p = 2, 3,4, 5 were chosen to define these
j=I
categories. We chose these categories in this way because empirical evidence seems to
indicate that the complexity of these problems is more sensitive to the magnitude of p
than to the magnitudes of k or n (Kuno, Yajima and Konno 1993). Within each
category, the test problems were classified into subcategories of 10 problems, each
defined by the values of the ordered pair (k,n).
To help evaluate the attractiveness of the solutions found by the heuristic
algorithm, we found a global optimal solution for each test problem by completely
enumerating all of the efficient extreme points of the associated multipleobjective linear
65
program (3.16). To accomplish this, we use the ADBASE computer code developed by
Steuer (1983).
Some statistics summarizing the results of these computations are presented in
Tables 3.13.4. In each table, each row gives average statistics for a subcategory (k,n) of
10 problems, a measure of the worst case performance of the heuristic, and the number of
problems in a category for which a global optimal solution was found. The first statistic is
the average number of efficient extreme points found by ADBASE in solving the
problems by complete enumeration. In some sense, the magnitudes of these numbers
correspond to the average relative difficulties, by subcategory, of each group of 10 linear
multiplicative programs in a subcategory. The second statistic is the average efficiency
rating r given by
r = 1 [(ZH Zin .)/(Z Z. )],
where z, is the objective function value returned by the heuristic, and where z,, and
z, are the global minimum and maximum values of the objective function of the test
problem over the corresponding set of efficient extreme points of (3.16). Thus, 0
and the closer r is to 1.0, the more attractive the value z, returned by the heuristic is
relative to the actual global minimum value zn, The third statistic given for each
subcategory in these tables is the average CPU time (seconds) that the heuristic needed to
solve a problem in the subcategory. The fourth statistic shows the lowest efficiency rating
calculated for a problem in the subcategory. It gives a measure of the worst case
performance of the heuristic algorithm when applied to the 10 problems in a subcategory.
66
Table 3.1. Computational Results: p = 2.
Subcategory Avg No. Avg. Eff. Avg. Solutions Lowest Eff. No. Exact
k n Eff. Points Rating r Time (sec.) Rating r Solutions
25 20 28.8 1.000 0.227 1.000 10
25 30 28.8 1.000 0.241 1.000 10
30 40 47.9 1.000 0.389 1.000 10
40 30 28.2 1.000 0.328 1.000 10
40 50 47.0 0.999 0.504 0.996 8
50 40 35.1 0.999 0.453 0.999 9
50 60 29.2 1.000 0.556 1.000 10
60 70 62.3 1.000 1.070 1.000 10
The fifth statistic is the number of problems in a category for which the heuristic
algorithm found a global optimal solution.
These four tables show that the solutions returned by the heuristic algorithm give,
on the average, quite accurate estimates of the actual global minimum values for the 260
linear multiplicative test problems generated. This is indicated by the fact that average
efficiency ratings by subcategory always were at least 0.920, and in approximately 96%
of the subcategories exceeded 0.950. It is noteworthy that, for these problems, these
ratings r by subcategory do not seem to decline significantly as p, k, and n increase in
Table 3.2. Computational Results: p = 3.
Subcategory Avg. No. Eff. Avg Eff. Avg. Solutions Lowest Eff. No. Exact
k n Ext. Points Rating r Time (sec.) Rating r Solutions
25 20 330.6 0.985 0.321 0.951 4
25 30 896.8 0.960 0.469 0.708 5
30 40 873.3 0.987 0.543 0.884 7
40 30 949.3 0.993 0.609 0.968 6
40 50 2073.7 0.920 0.967 0.806 4
50 40 1484.9 0.993 0.908 0.961 7
50 60 2846.3 0.995 1.298 0.978 6
60 70 5867.5 0.969 2.495 0.799 2
67
Table 3.3. Computational Results: p = 4.
Subcategory Avg. No. Eff. Avg Eff. Avg. Solutions Lowest Eff. No. Exact
k n Ext. Points Rating r Time (sec.) Rating r Solutions
25 20 2789.5 0.998 0.426 0.993 4
25 30 7245.9 0.992 0.598 0.945 5
30 40 23656 0.986 1.019 0.947 1
40 30 19034 0.978 0.998 0.923 2
40 50 50889 0.969 1.539 0.918 0
50 40 59443 0.969 1.587 0.843 2
50 50 83780 0.981 1.901 0.890 3
value. In addition, with the exception of one subcategory, a global optimal solution was
found for at least one problem in each subcategory.
The average solution times by subcategories shown in the four tables indicate that,
for these test problems, the computational effort required by the heuristic was rather
small. In fact, these average times were always less than 2.50 seconds. In comparison to
exact algorithms that have been used in test situations to globally solve linear
multiplicative problems, these times are generally either at least as small or much smaller
(see, e.g., Kuno, Yajima, Konno 1993 and Ryoo and Sahinidis 1996). Furthermore, in
contrast to solution times for exact algorithms, these average solution times seem much
less sensitive to increases in p, n, k or to increases in the average number of efficient
Table 3.4. Computational Results: p = 5.
Subcategory Avg. No. Eff. Avg. Eff. Avg. Solutions Lowest Eff. No. Exact
k n Ext. Points Rating r Time (sec.) Rating r Solutions
10 20 1331.4 0.993 0.353 0.941 5
20 10 527.1 0.998 0.294 0.993 2
25 30 57115 0.995 0.962 0.992 2
68
extreme points that exist in the corresponding problems (3.16); see Kuno, Yajima, Konno
(1993) and Ryoo and Sahinidis, (1996).
Finally, it is worth noting that we were able to apply the heuristic to much larger
problems than those reported in Tables 3.13.4. However, the number of efficient extreme
points in the associated multipleobjective linear programming problems (3.16) for these
cases always exceeded 200,000. Since the ADBASE code cannot be used to find all of the
efficient extreme points for such problems, we were unable to completely enumerate the
sets of efficient extreme points to find z,, and r values for these problems. Thus, we are
as yet not able to draw conclusions concerning the accuracy of the heuristic for any
problems larger than those reported in Tables 3.13.4.
To further illustrate the effectiveness in practice of the heuristic algorithm, we
solved a real application problem in forest management that was studied in Steuer and
Schuler (1978) as a multipleobjective linear programming problem. The problem
involves the allocation of land and budget monies in a way that seeks to maximize
objectives in timber production, hunting and cattle grazing in the Swan Creek subunit of
the Mark Twain National Forest. Steuer and Schuler (1978) provide actual data used to
formulate their multipleobjective linear programming problem. The problem contains 31
decision variables, 5 linear objective functions, and 13 constraints. Our multiplicative
programming problem was formed from this problem by multiplying the 5 linear
objective functions together to form a single objective function. The heuristic was then
used to search for an approximate solution that maximizes this single objective function
subject to the constraints of the forest management multipleobjective linear
programming problem.
To help evaluate the attractiveness of the solution found by the heuristic
algorithm, we found a global optimal solution by enumerating the 83 efficient extreme
points of the associated forest management multipleobjective linear program using the
ADBASE computer code. An efficiency rating of r = 0.999 was calculated using the
slightly modified equation
r= 1 I Z (z,, )]
since this multiplicative programming problem is a maximization problem rather than a
minimization problem. This efficiency rating indicates that the heuristic algorithm
returned an attractive value z, relative to the actual global maximum value z,.
3.5. Discussion
The results of this chapter imply that there are at least two ways to rewrite a
concave multiplicative programming problem as a concave minimization problem. It
follows that concave minimization theory and methods can be used in these ways to
analyze and solve concave multiplicative programs. The results also imply that a concave
multiplicative programming problem can be analyzed and solved directly, without any
reformulation, as a quasiconcave minimization problem over a convex set. Furthermore,
the analysis in the chapter implies that any concave multiplicative programming problem
(Px) with a compact feasible region has at least one optimal solution that is an efficient
extreme point solution of the associated multipleobjective mathematical programming
problem (3.15). Therefore, the opportunity exists for devising solution methods for such
problems (Px) that search among the efficient extreme points of the associated multiple
objective problems (3.15). The chapter proposes a heuristic algorithm that takes this
70
approach for solving linear multiplicative programs. From the computational results
presented for this heuristic algorithm, we conclude that its features and performance offer
significant potential for conveniently finding very attractive solutions with relatively little
computational effort to the various applications using linear multiplicative programming
encountered in practice. Thus, the theoretical and algorithmic results presented in this
chapter offer some potential new avenues for more effectively analyzing and solving
multiplicative programming problems of various types.
CHAPTER 4
A GENERAL MULTIPLICATIVE PROGRAMMING PROBLEM IN OUTCOME
SPACE
4.1. Introduction
Recall from Chapter 1 that the multiplicative programming problem is given by
(Px) vx =min fl(x),s.t.xe X,
where p > 2 is an integer, X is a nonempty set in R", and, for each j = 1, 2,..., p,
f,: X 4 R satisfies f,(x)> 0 for all xe X. For simplicity, we assume that the
minimum v, in problem (Px) is achieved.
For any xe R", let f(x) denote the pvector withjth entry equal to f,(x),
j = 1, 2,..., p. Let ye RP denote the pvector withjth entry equal to yj, j = 2,..., p.
For each j = 2,..., p, let iY e R satisfy
5j1sup f,(x),s.t.xe X,
where 5j = +o is possible, and let e RP denote the vector with jth entry equal to y,
j = 1, 2,..., p. Although various outcomespace reformulations of problem (Px) have
been proposed for solution purposes, one of the most common reformulations is given by
the problem
(P') v, = min g(y), s.t. ye Y4,
where
YS ={yE RPIf(x)yg forsomexe X, (4.1)
and where, for each ye Y g : Y + R is defined by
g(y) = l Y. (4.2)
j=1
For example, problem (P, ) is essentially the reformulation of problem (Px) used in the
algorithms of Benson (1998c), Falk and Palocsay (1994) and Thoai (1991). Notice that
since X is nonempty, Ys is a nonempty set. By constructing appropriate global solution
algorithms for problem (P,r), this problem provides us with the opportunity to solve
problem (Px) by working in the outcomespace RP of the problem, rather than in the
decision space R", which is generally much larger than RP. In order to globally solve
problem (P,'), it is important to understand the properties of the set Ys defined by (4.1),
of the function g defined by (4.2), and of problem (P, ) itself.
This chapter undertakes a mathematical analysis of the outcomespace
reformulation (P,,) of problem (Px). The analysis is organized according to whether or
not the outcomespace problem satisfies conditions for the general case, the convex case,
or the polyhedral case. For the general case, we show, for instance, that globally solving
either problem (Px) or problem (P J) essentially also globally solves the other problem,
and that, for any feasible point y for problem (Pr,), either g(y)< g(y) for some ye Y
or y satisfies a condition that is necessary, but not sufficient, for it to be a local optimal
solution for problem (P,"). For the convex and polyhedral cases, we show stronger
73
results. For example, we show for the convex case that any global optimal solution for
problem (P,) must lie on the boundary of YV, that the objective function g in problem
(P_,) is strictly pseudoconcave on Y, and, when YV is closed and contains at least one
extreme point, that problem (P,,) has an extreme point global optimal solution.
The analysis of the general case of problem (P,) is given in Section 4.2. Section
4.3 provides analytical results for both the convex and polyhedral cases of problem (Py_).
4.2. Results for the General Case of Problem (P,,)
Notice under the assumptions made in Section 4.1 for problem (Px) that Y is a
nonempty subset of RP := z e R'Pz > O0. When Y satisfies this condition, we obtain
what we will call the general case of problem (P,,).
It is important to establish that by solving the general case outcomespace
formulation (PF,) of problem (Px), a global optimal solution for problem (Px) can be
recovered. The following result, by showing that problems (Px) and (P.) are equivalent
in a certain sense, immediately establishes this fact.
Theorem 4.2.1. (a) If x' is a global optimal solution for problem (Px), then y = f(x*)
is a global optimal solution for problem (P,). Furthermore, v, = v,.
(b) Problem (P,,) has at least one global optimal solution. Furthermore,
if y' is a global optimal solution for problem (Py,), then any x' e X such that
f(x' ) y* is a global optimal solution for problem (Px).
74
Proof. (a) Let x' be a global optimal solution for problem (Px), and set
y' = f(x'). From (4.1) and (4.2), this implies that y" e Y and that
g(y')= f f(x)=v,.
j=1
Therefore, v,
there would exist an xe X such that
o< lf,(T) g(y)< .
11
which contradicts the definition of v,. Therefore, g(y)v, for all ye YV. This implies
that v, v,. Since v,
a global optimal solution for problem (Pr,).
(b) By assumption, we may choose a global optimal solution for problem
(Px). From part (a), this implies that problem (P,) has at least one global optimal
solution. Suppose that y' is a global optimal solution for problem (Pr,). Since y' e YV,
(4.1) implies that we may choose an arbitrary x' e X such that f(x' ) y'. Then, from
(4.2), since 0
Since x' e X and y' is a global optimal solution for problem (P_), this implies that
vY~ f, (x')
jil
75
From part (a), v, = v,. By (4.3), this implies that If,(x'*)= v. Since x' e X, it follows
j=I
that x' is a global optimal solution for problem (Px). O
Suppose in the general case of problem (Py,) that a point ye Y has been
generated. For algorithmic purposes, it may be valuable to have a tool for finding an
alternate point ye Y that satisfies g(y)< g(y), if such a point exists. The next result
gives an idea for potentially helping to create such a tool. To prove this result, we need
the following lemma. This lemma will also be useful in proving several other results later
in this chapter.
Lemma 4.2.1. Assume that E YV. Then, for any yE Y',
(l/p)(Vg(y), y) = g(y)(l p) i (y/y),
j=l
and
g(yM)/Ap) (/j y gg(yj)g(y)g g(y)
jI=
with equality holding in the latter relationship iff, for some constant M > 0, y, = M y,
j=l, 2,..., p.
Proof. Choose an arbitrary point ye Y. Suppose that ye YV. Then, by (4.2),
since Y' cR, g(y)> 0. By definition of g,
(1/p)(Vg(y),'y)= (l/p) ky j
= (i/p)[g(y)/Yj ]Y
j=I
76
=g(y) (1/p)(y /y). (4.4)
j=1
Since (l/p)>0, (yj/y)> 0 for each j = 1, 2,...,p, and p(1/p)= 1, the arithmetic
geometric mean inequality (Duffin, Peterson, and Zener 1967) implies that
(i/P)(y/YJ) g(Y)/8(Y),
j=I
with equality holding iff for some constant M > 0, y, = My for each j = 1, 2,..., p.
Together with (4.4), since g(y)>0, this implies the desired results. [
Theorem 4.2.2. Assume that ye Y. If
1.0 > inf(l/p) (y,/y), (4.5)
then g(y)< g(y) for some yE Y. In particular, if y achieves the infimum in (4.5), then
g(y)< g(y).
Proof. Suppose that y e Y. If (4.5) holds, then for some e YV,
1.o>(/p)(y,/y,). (4.6)
J=i
Since g(y)> 0, this implies that
g(y)> g(y)(1i/p) (/y,). (4.7)
j=l
/i
From Lemma 4.2.1, since y e Y', we know that
g(y)(l/p) (jt/yj ) g(y)[g (y)/g(y)]i/P. (4.8)
j=j
Since g(y)> 0, together (4.7) and (4.8) imply that
77
1.0> [g()/g(y)]/P.
Because g(y)> 0, this implies that g(y)< g(y). Therefore, g(y)< g(y) for some
ye YV. Since, for any point 5 that achieves the infimum in (4.5), (4.6) is also satisfied,
the argument above also implies that if y achieves the infimum in (4.5), then
g(y)< g(y).
Notice that when y e Y, the infimum in (4.5) is either less than 1.0 or equal to
1.0. From Theorem 4.2.2, when this infimum is less than 1.0, a point y in YV such that
g(y)< g(y) exists. In particular, in this case y is not a global optimal solution for
problem (P,). The next result covers the case when the infimum in (4.5) equals 1.0.
Theorem 4.2.3. Assume that ye Y If
1.0 = inf (/p)(,/y,), (4.9)
then y is an optimal solution to
vd = min(Vg(), y y), (4.10)
and vd =0.
Proof. From (4.9), since ye Y the infimum in (4.9) is achieved at y = y. By
Lemma 4.2.1, since g(y) is a positive constant, this implies that y also minimizes
(1lp)(Vg(y), y) over Y'. Since (l/p) is a positive constant and (Vg(y), y) is a
constant, it is easy to see that this implies that y is an optimal solution to (4.10) and
d =0. ]
78
A point ye Y is a local optimal solution for problem (P,,) when there exists an
E >0 such that for each ye Y for which Ilyy le, g(y)g(y). From Theorem 4.2.3,
when ye Y and (4.9) holds, then, for any y E Y, if there is a S > 0 such that
d := (y y) satisfies y + Ad e Y for all A such that 0 < A56, the directional derivative
of g at y in the direction d will be nonnegative, i.e., (Vg(y)d)0. From Bazaraa,
Sherali and Shetty (1993), this is a necessary, but not sufficient condition for y to be a
local (or global) optimal solution for problem (P,,).
4.3. Results for Convex and Polyhedral Cases of Problem (PY,)
When Y, in addition to being a nonempty subset of RP, is a convex set, then we
obtain what we will call the convex case of problem (P,). Similarly, when Y, in
addition to being a nonempty subset of RP, is a polyhedron, then we obtain what we will
call the polyhedral case of problem (P,). Each of these types of outcomespace versions
of problem (Px) arises from a broad class of decision space problems, as shown by the
next result.
Theorem 4.3.1. When X is a convex set and, for each j = 1, 2,..., p, f is a convex
function on X, we obtain the convex case of problem (Pr). When X is a polyhedron
and, for each j = 1, 2,..., p, f, is linear on R", we obtain the polyhedral case of
problem (PY,).
Proof. Assume, in addition to the assumptions made in Section 4.1 on X and on
fj, j= 1, 2,..., p, that X is a convex set and that, for each j= 1, 2,..., p, f, is a
79
convex function on X. We will show that Y is a convex set. Choose any y', y2 E Y.
From (4.1), since y', y2 e Y, we may choose x', x2e X such that f (x' ) yl and
f,(x2)
(1A)>0, for each j=l, 2,..., p,
Af, (x)+ (IA)f,(x2)Ay' + ()y2. (4.11)
By the convexity of f,, j = 2,..., p, on the convex set X, if we set x= x' +
(1 )x2, then
xe X, (4.12)
and, for each j = 1, 2,...,p,
f ,()A f,(xl)X )+(1)f, (x). (4.13)
From (4.11)(4.13), f (x)) A y' +(1 )y2, where e X Since y', y2 Y, y' Y
holds for each i = 1, 2. As a result, since A, (1 .A) 0, A y' + (1 A)y2 Y. The
conditions for A y' + (1 A)y2 to belong to YV are thus satisfied. By the choices of y', y2
and A, this implies that Y is a convex set.
Now suppose, in addition to the assumptions made in Section 4.1 on X and on
f, j = 1, 2,..., p, that X is a polyhedron and that, for each j = 1, 2,..., p, fj is a linear
function on R". We will show that Y is a polyhedron. By definition, since X is a
polyhedron, there exists a finite number q of linear functions g,, j = 1, 2,...,q, on R",
and real numbers b j = 1, 2,..., q, such that
X = xe R"lg (x)b,, j=1,2,...,q}.
80
Let Z F R""+ be defined as the set of all solutions (x, y) to the system of linear
inequalities (4.14)(4.16) given by
f,(x) Y<0, j=1, 2..., p, (4.14)
yj.j, j=l,2,...,p, (4.15)
g,(x)
Then, by definition, Z is a polyhedron in R" Let A be the p x (n+ p) matrix whose
first n columns each equal 0e RP and whose last p columns together form the px p
identity matrix. Then, from (4.1) and the definition of Z, Y = AZ. From Rockafellar
(1970, Theorem 19.3), YV is a polyhedron in R.. O
In convex cases of problem (Pr,) (and thus, in polyhedral cases as well), certain
locations within Y for seeking global optimal solutions can be specified. For instance,
we have the following result.
Theorem 4.3.2. Suppose that problem (P,,) satisfies the conditions for the convex case.
Then:
(a) Any global optimal solution for problem (P,) belongs to the boundary of YV.
(b) If YV is closed and contains at least one extreme point, then there exists at
least one global optimal solution for problem (Pr,) that is an extreme point of YV.
Proof. Assume that YV, in addition to being a nonempty subset of R.P, is a
convex set, i.e., that we have the convex case for problem (P r). Then, from Theorem
4.2.1, problem (P,,) has at least one global optimal solution.
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(a) To show this part of the theorem, let y' denote an arbitrary global optimal
solution to problem (Pr,). Suppose that y' is not on the boundary of YV. By the choice
of y* and since Y5 is a convex set, Y has a nonempty interior. Therefore, y' must
belong to the interior of Y". From (4.1), this implies that for some xe X, f(x)< y'
must hold. By assumption, since X, f(xI)> 0. Therefore, if we set y = f (), it
follows that ye Y and
YIj < YI;
j=I j=l
From (4.2), this contradicts the global optimality of y' in problem (P.,). Therefore, y
must belong to the boundary of Y .
(b) From the discussion in Section 3.2, since Y is a nonempty convex set and,
for each j = 1, 2..., p, the function h (y) = y is positive and concave on Y, the global
optimal solution set for problem (P,,) is identical to the global optimal solution set for
the problem
(Pr) min (y),s.t.ye Y,
where, for each y e Y : Y R is the concave function defined by
(Y)= Y p.
Since YV is a nonempty, closed convex set with at least one extreme point, from
Rockafellar (1970, Corollary 18.5.3), it is easy to see that Y can contain no lines.
Furthermore, since problem (P,) has at least one global optimal solution, problem (P s)
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also has at least one global optimal solution. By Rockafellar (1970, Corollary 32.3.1),
since k is a concave function on Y, the latter two statements imply that problem (P,)
has at least one global optimal solution y that is an extreme point of Y'. Because the
optimal solution sets of problems (P,,) and (P,3) coincide, this completes the proof. [
Suppose that Ys is a nonempty, closed convex subset of Rf, and that Ys
contains at least one extreme point. Then, from Theorem 4.3.2, there will exist at least
one global optimal solution for problem (P,,) that is an extreme point of Y, and all
global optimal solutions for problem (Pr,) will lie on the boundary of Y. Neither of
these properties, however, is necessarily shared by the decision setbased problem (Px)
whose outcomespace reformulation yields problem (P,_). The following example
demonstrates this.
Example 4.3.1.Let p = 2, X = {(x.,x2f E R210x, S6, i = 1, 2,
f,(x1,x2)= ( 1)2 +1,
and
f,(x.,x,)=(x2 2) +
in problem (Px). Then X is a nonempty, convex set and for each i = 1, 2, f, is a
convex, positivelyvalued function on X Therefore, by Theorem 4.3.1, the problem
(Py,) obtained by formulating the outcomespace version of problem (Px) is guaranteed
to satisfy the conditions of the convex case for problem (Ps,). Furthermore, it is not
difficult to show, in this case, that Y is compact. Thus, Y is closed and contains at
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least one extreme point. It is easy to see that the unique global optimal solution to
problem (P _) is (y ) = (1, 1) which, as guaranteed by Theorem 4.3.2, is an extreme
point of Y (and is thus on the boundary of Y ). On the other hand, the only global
optimal solution to problem (Px) is (x'J = (1, 2), yet x' is neither on the boundary of
X nor is it an extreme point of X.
To present the next result, we need to define two types of functions.
Definition 4.3.1. Let Z c R" be a nonempty convex set, and let h: Z > R. The function
h is said to be quasiconcave on Z when for each z', z2 Z and A R such that
0
h[iz' + (l )z2 ]> min h(z h(z2)}.
Definition 4.3.2. Let W be an open set in R" that contains Z R R", and let h: W > R.
The function h is said to be strictly pseudoconcave over Z when h is differentiable over
Z and, for each distinct z', z E Z, if (Vh(z'), z2 zl) O, then h(z2)< h(z').
It is well known that a differentiable, quasiconcave function h: Z  R need not
be strictly pseudoconcave over Z. For a discussion of quasiconcave and strictly
pseudoconcave functions see, for example, Bazaraa, Sherali and Shetty (1993).
From Konno and Kuno (1995, p. 379), we know that when Y is a convex set,
since Y5 C RP, g: Y< 4 R defined by (4.2) is quasiconcave on Y4. Thus, in the convex
case, problem (P,) is a minimization of a quasiconcave function over a convex set. In
fact, however, we have the following even stronger result.
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Theorem 4.3.3. Suppose that problem (P,,) satisfies the conditions for the convex case.
Then, in this problem, g is a strictly pseudoconcave function over the convex set Y.
Proof. The set Y is a convex set by definition of the convex case for problem
(P,,). To show that g is strictly pseudoconcave over Y, notice first that by (4.2), g
can be considered to be well defined over the open set RP. Also notice that g is
differentiable over RP and, thus, over Y g RBf.
Suppose now that y' and y2 are distinct points in Y" that satisfy
(Vg(y'), y2 y'l 0. Then, from (4.2), we obtain
O >(Vg(y),Y2 yl)= Y I Y)
k=1 j k
= Y Y y g(y)
k=l jl k=1
= [g(y' )](Y/y: ) pg(y')
k=1
= g (y' )[(y /y) p]. (4.17)
By multiplying both sides of (4.17) by (l/p) and rearranging, we obtain that
g(y ) g(y')(1/p) (y/y: ). (4.18)
k=i
From Lemma 4.2.1,
gy')(1/p) y/y: ) g(yl )[g(y2 l (4.19)
ii
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with equality holding iff, for some M > 0, y2 = M y[, k = 1, 2,..., p. There are two
cases to consider.
Case (i): There is no M >0 such that y = Myk, k = 2,..., p. Then, in (4.19),
strict inequality holds, so that from (4.18) and (4.19),
g(y')> g(yI [g(y2g(y )]'/
Since g(y')>0, this implies that g(y)< g(y').
Case (ii): For some M > 0, y2 = M y, k = 1, 2,..., p. If we choose such an M,
then (4.19) holds as an equality. Thus, from (4.19) and the choice of M, we obtain that
g(y' )(1/p)(y/y: )= gy' )[g(y2 ) g(y' )f/ (4.20)
k=1
and that
g(y2)=Mpg(y), (4.21)
respectively. Since g(y')> 0, together (4.18), (4.20) and (4.21) imply that
g(y)' gG XM P) = MgGy
Dividing through by g(y')> 0 yields M <1. Notice that M # 1, since, by assumption, y'
and y2 are distinct. Therefore M <1. By (4.21), since g(y'), g(y2)> 0, this implies that
g(y2)< g(y'), and the proof is complete. [
Remark 4.3.1. Theorem 4.3.3 justifies and strengthens the claim of Sniedovich and
Findlay (1995, p. 317) that when YV is a convex subset of Rf, g: Y ) R defined by
(4.2) is differentiable and pseudoconcave on Y.
86
From Theorem 4.3.3, in the convex case, problem (P,) is a global optimization
problem involving the minimization of a strictly pseudoconcave function over a convex
set Y. Therefore, as in the general case, multiple local optimal solutions for problem
(P,,) will generally exist that are not globally optimal.
From Theorem 3.2.1, we know that when Y is a nonempty, convex subset of
RP, the function : Y< 4 R defined, as in the proof of Theorem 4.3.2, by
R(y)= [g(y)lp (4.22)
is concave where g : Y' R is given by (2). By the next result, when the domain of g is
restricted to an appropriate subset of Y, a stronger statement can be made.
Theorem 4.3.4. Assume that Y is a nonempty, compact, convex subset of R.P. For any
ae R' and be R such that a >0 and b>0, let Z(a,b)=Y n {ye RPl(a,y)=b}.
Then R : Z(a,b)  R defined for each y E Z(a, b) by (4.22) is a strictly concave function
for any a e RP and be R such that a > 0 and b > 0.
Proof. Assume that yl,y2 EZ(a,b) and y' y2, where ae R', be R, a>0,
and b > 0. Since Z(a, b) is an intersection of two convex sets, it is itself a convex set.
Therefore, if we choose A E R such that 0 < A < 1, then
z:=Ay'+(lA)y E Z(a,b).
Also, by (2) and (4.22),
(z)=A [Ay' +(1)yJl P. (4.23)
L/='
From Polya and Szego (1972),
87
Ay+(1 A)y+j] s y + (i Y y) ] (4.24)
j=1i J { = j=1 I J=1
with equality holding iff A y) = K(l )y', j= 1, 2,.... p, for some positive constant K.
Since
and
[(I~ y (lA1 (y2)
(4.23) and (4.24) will imply the desired result if we can show that no K > 0 exists such
that
A.y =K(1AI)y', j=1,2,...,p. (4.25)
Notice that since y' f y2, Kx := [A/(l )] does not satisfy (4.25).
Suppose, to the contrary, that for some K > 0, (4.25) is satisfied. Then from
(4.25) it follows that
y'= K[(1 )/y]y2. (4.26)
Since y', y2 e Z(a,b),
(a, yl= (a, y2 =b. (4.27)
Substituting for y' in (4.27) via (4.26), we obtain
K[(l )/A;(a, y) =(a, y2)=b.
88
Solving here for K, we obtain that K = [/(1 A)]. Since K = K, = [A/(1 A)] does not
satisfy (4.25), this contradiction concludes the proof. O
It is important to notice that the counterpart of Theorem 4.3.4 in the decision
space does not hold, even in the polyhedral case. In particular, suppose that X C R" is a
nonempty, compact polyhedron and, for each j = 1, 2,..., p, that there exists a c, a R"
such that f (x)= (ci, x) >0 for all xe X. Then, although the function h: X R
defined for each x eX by
h(x)=l (c, x (4.28)
is concave (see Theorem 3.2.1), the function h: X(a,b) R need not be strictly
concave, where aE RP, be R, a>0, b>0, and
X(a,b)= xeX ac,,x)=b}.
The following example illustrates this observation.
Example 4.3.2. Let
X ={(x,,x2 e R: 0 5 <, 4.0,i=1,2},
and let f,(x,,x,)= ((1, l),(x,, x)), j= 1, 2. Then X is a nonempty, compact polyhedron
and, for each j = 1, 2, f, is positive and linear on X. As guaranteed by Theorem 3.2.1,
h: X  R, which, by (4.28), is given by
h(xi,,x)=(x, +x2),
89
is concave. However, if, for example, a, = a2 = 1 and b = 4, then h is not strictly
concave on
X(a,b)= (x,,x2,0l.5x,< 4.0,i= l,2,x, +x2 =4.
Consider now problem (Pr,) when the conditions of the polyhedral case hold.
Assume also that Y is a compact set, and that ye Y. For algorithmic purposes, it may
be quite useful in this case to develop tools for finding local optimal solutions for
problem (PY). These tools could then potentially be used to construct global solution
algorithms for the problem that repeatedly move from a local optimal solution to an
improved local optimal solution until a global optimal solution is found. The remaining
results in this section are motivated, in part, by the desire to find such tools.
Notice that in the polyhedral case, the optimization problem in (9) is a linear
program given by
(LP) min (1/p)~ (yj/), s.t. ye y .
j=1
Problem (LP) will have an optimal solution y' that can be found, for instance, by the
simplex method. Since ye Ys, the minimum value vmin in problem (LP) satisfies vmin
S1.0. As a result, there are three possible cases for problem (LP). First, vmin <1.0 may
hold. Second, vmin = 1.0 may hold, with y being the unique optimal solution to problem
(LP). Third, vmin = 1.0 may hold, with problem (LP) having multiple optimal solutions.
In the first case, from Theorem 4.2.2, it follows that gy')< g(y), where y* is any
optimal solution to problem (LP), so that a more attractive feasible solution y" to
90
problem (P,) than y has been found. To analyze the second case, we need the
following two definitions and lemma.
Definition 4.3.3. A point ye Y is a strict local optimal solution for problem (P,,) when
there exists an E > 0 such that for each ye Y for which y y and ly I < e,
g(y)> g(y).
Definition 4.3.4. Let Z be a nonempty convex set in R', and let h: Z * R. The function
h is said to be strongly quasiconcave on Z when for each z', z2 e Z with z' z2, we
have
h[A z' +(1A 2]> min (z' ),h(z2)
for each A such that 0 < ; < 1.
Lemma 4.3.1. Let Z be a nonempty convex set in R", and let h: Z * R be strongly
quasiconcave. Suppose that z', i= 1, 2...,k, are distinct points in Z and that s is an
element of the convex hull of z', i = 1, 2,...k, such that, for each i = 1, 2,...,k, s z'.
Then
h(s)> min h(z')i= l,2...k }.
Proof. The lemma is easy to prove using Definition 4.3.4 and induction. O
The following result analyzes the case where vmin = 1.0 and y is the unique optimal
solution to problem (LP).
Theorem 4.3.5. Assume that problem (P',) satisfies the conditions for the polyhedral
case, and that YV is a compact set. Assume also that ye Y. Suppose that vmin = 1.0
91
and that y = y is the unique optimal solution to problem (LP). Then y is a strict local
optimal solution for problem (Pr').
Proof. Since g(y)> 0 and y = y is the unique optimal solution to the problem
min(l/p) (y,/ly,)
Y.Y j=1
y = y must also be the unique optimal solution to the problem
mi g(y)(l/p) (yj/G ).
'Y6 j=l
Therefore, by Lemma 4.2.1, y = y is the unique optimal solution to the problem
min(l/p)(Vg(Y3 y).
yEY'
Since (l/p)> 0 and (Vg(y), y) is a constant, this implies that y = y is the unique
optimal solution to the problem
min(Vg(y), y y). (4.29)
eVY
Therefore, the optimal value of problem (4.29) is 0, and for all ye Y such that y ; y,
(Vg(y),y y) > 0. (4.30)
Let d', d2...,dk represent the directions of the edges of Y5 emanating from the
extreme point y of Y. From (4.30),
(Vg(y),d')>0
for all i= 1, 2,...,k. By Theorem 4.1.2 in Bazaraa, Sherali, and Shetty (1993), this implies
that there exist positive reals 6,, i = 1, 2,...,k, such that
g(yG+ d')> g(y) (4.31)
92
for each Ae (0,6,). Let 8= 1/2 min { i = 1, 2,...,k}, and consider the points y and
y+6d', y+&d2 ..., y+.d.
Then by definition of 8 and (4.31),
g( + d')> g(y) (4.32)
for each i = 1, 2,...,k. Let z be any element of the convex hull of y, y +&d, i= 1, 2,
...,k, such that z # y and, for each i= 1, 2,...,k, z Y+ y d'. Since g is a strictly
pseudoconcave function on Y, it is also a strongly quasiconcave function on Y
(Bazaraa, Sherali, Shetty 1993). As a result, by Lemma 4.3.1,
g(z)> min (y) g(y +&d'), i = 1, 2,...,k}. (4.33)
From (4.32) and (4.33), g(z)> g(y). Since 8 >0, this implies that there exists an e >0
sufficiently small so that if ze Y', IIz y< e, and z Y, then g(z)> g(y). ]
Under the assumptions of Theorem 4.3.5, if vmin = 1.0 but y = y is one of two or
more optimal solutions to problem (LP), then y need not be a strict local optimal
solution for problem (P, ). The following example illustrates this point.
Example 4.3.3. Let
Y = (y, E R21y + y, >8, y1 7, y2 4j,
and let yT = (4, 4). Then Y is a nonempty, compact polyhedron in R2, and the
assumptions of Theorem 4.3.5 are satisfied. In this case, ye Y6 and y is an optimal
solution to problem (LP). However, since (y~ = (4 +, 4 8 e Y' and g(y6)< g(y)
for all values of 6 such that 0 < < 3, by Definition 4.3.3, y is not a strict local optimal
93
solution for problem (P,,). (In fact, y is not even a local optimal solution for problem
(Pr,).) Notice that problem (LP) in this case has multiple optimal solutions.
In the third case of problem (LP), vmin = 1.0 and problem (LP) has multiple
optimal solutions. In this case, by the next result, as in the first case, an improved feasible
solution for problem (Py,) is at hand. The proof of this result relies crucially on Theorem
4.3.4.
Theorem 4.3.6. Assume that problem (Pr,) satisfies the conditions for the polyhedral
case, and that Ys is compact. Suppose that y is an optimal solution for problem (LP),
and suppose that problem (LP) has multiple optimal solutions. Then, for any y' # y that
is an optimal solution for problem (LP), g(y) < g(y) must hold.
Proof. Let y* # y be an optimal solution to problem (LP). Then, since g(y)>0,
y' is also an optimal solution for the problem
min g(y)(1/p)(y /y,), s.t. ye YV.
By Lemma 4.2.1, since y YV, this implies that y* is an optimal solution to the problem
min (1/p)(Vg(y), y), s.t. ye Y".
Since (l/p)(Vg(y), y) is a fixed number, it follows that y' is an optimal solution to the
problem
min (l/p)(Vg(y), y), s.t. ye Y. (4.34)
By assumption, y is an optimal solution for problem (LP). Therefore, the optimal value
of problem (LP) equals 1.0. From Theorem 4.2.3, this implies that the optimal value of

Full Text 
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MULTIPLICATIVE PROGRAMMING: THEORY AND ALGORITHMS By GEORGE BOGER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999
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ACKNOWLEDGMENTS I would like to thank my entire supervisory committee Dr. Harold Benson, Dr. Selcuk Erenguc, Dr. Asoo Vakharia, and Dr. Richard Francis for their time and helpful feedback on my dissertation I am especially grateful to my committee chairman, Dr. Benson, for suggesting the topic of multiplicative programming problems and for his tremendous assistance and unending support. Without his help, this dissertation would not have been completed. I would also like to thank Mr. Erijang Sun for proving some theoretical results needed to support my dissertation topic I am also grateful to the DIS department chairman, Dr. Erenguc, for providing an assistantship and for allowing me to teach undergraduate courses during my time at the University of Florida The teaching experience was an enjoyable and rewarding expenence. I would like to thank my family for their encouragement and emotional support. I would also like to thank my colleagues in the Ph.D. program for their friendship and their support. Finally, I am in debt to my master's degree advisor, Dr. Frederick Buoni, at the Florida Institute of Technology, for his guidance. He suggested multiple objective linear programming as a topic for my thesis. While working on the thesis, I met Dr Benson during a visit to FIT to present a talk related to multiple objective linear programming. 11
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Dr. Benson agreed to serve on my master's degree committee and later recruited me for the DIS Ph.D. program. 111
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TABLE OF CONTENTS page S .. AC.KN' 0 WLEDG MENT . ... ........ .. ..... ........ .. .... . ... .... .. . ..... .... .. .... . .. . . ... ................ 11 ABSTRACT ....... ......... .. . .. ...... .. ....... .. .. ..... .. .. . .... ....... .............. .... .. ........ ... ........ .. ... Vl CHAPTERS 1 rnTRODUCTION .. .. ..... ................ ... .. ... . .. ....... ... .. . .......... .... .... ....... ...... .... .... l 1.1. The Multiplicative Programming Problem ........... ........ ..... .. . .. ... .. ...... ... .. 1 1.2 Reformulation s of the Multjplicative Programming Problem . ... ......... ..... 4 1 .3. Purpo se and Or ga nization of the Di sse rtation . . ........ . .. .... ....... . ... . .. .... .. 6 2 A REVIEW OF THE Lfl'ERATURE ON MUL'l'IPLICATIVE PROGRAM.M.IN'G PROBLEMS .. .. .. . .. ..... .. .. .. . ..... .... ... .... .. . ...... ..... 9 2.1. Or ga ni za tion of the Literature Review . .. .. .. .. ...... ...... .. . .. .... . . .. ......... .. . .... 9 2.2. Method s to Solve Problem s (LMP2), (GLMP), and (CLMP) . .. ... ...... ..... 13 2 .2. 1 Method s Based on Quadratic Programming ............ . .......... ........ 15 2.2.2. Method s Ba s ed on Searching the Outcome Set . .. . .. .... ...... ... ...... 17 2.2.3. Method s Based on Solving a Parametric Master Problem .. ........... 22 2.2.4. Method s Based on Polyhedral Annexation .. .... .... .. .... ............... . 28 2 .3. Extensions of Algorithms for Problem (LMP2) to Solve Problem (L MP) when p > 3 .... .. ........ ........ ....... .. ..... . .. ... . .. .. .. .... .. ........ . .... 32 2.4. M e th ods to Sol ve Problems (C MP ), ( GCMP) and (CCMP) . ........ .......... 32 2.4.1. Method s Ba se d on Solving a Reformulated Problem ..................... 33 2.4.2. A Method Based on Outer Approximation .............. .. . ... .... . . . .. .. 37 2.5. M e thod s to Sol ve Problem (LMP) as a Concave Minimization Problem ............................................... .. .......... . .. . .. ........... .... ........... 38 3 CONCAVE MULTIPLICATIVE PROG G PROBLEMS : ANALYSIS AND AN EFFICIENT PO.IN'T SEARCH HEURISTIC FOR THE L.IN'EAR CASE ..... .. ... .. . . .................................................... . 40 3.1. Introdt1ction .. .. .. .. .. . .. .. ..... .. .. . . .. .. ....................................... .... ........ ... 40 3 .2 Analysis .. ..... ..... .. .. .. . .. ..... .. .. .......... .. . ...................... ...... ... ... .. ............. 41 3.3. Efficient Point Search Heuristic .. ....... .. ... .. . ....................... ... ... ............ 52 3.4. Computational Result s ......... . ........ .. .. ..... .. . ......... ...... ... . .. . .. ........... . 62 3 5. Di sc u ss ion ......... ... . .. .. . . . .. .... .. ......... . .. .... . ..... .. .. ..................... . .. . .. . .. 69 IV
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4 A GENERAL MULTIPLICATIVE PROG _.._ G PROBLEM IN OUTCOMESPACE .. .. ..... ........ . .. .. . .. .. . ... .... ........ ....... . .............. ... .... .. 71 4.1. Introduction ...... . ...... .. ... ... .. .. .......... .. . ............. .. ..... .............. ...... . .......... 71 4 .2 Re s ult s for the General Case of Problem (Py s ) .... ............. ....... .... . ... .. .. 73 4.3 R es ult s for Convex and Polyhedral Cases of Problem (P s) .... . ........... .. 78 Y4 .4. Di sc u ss ion .............................. .... .. ..... ....... .. .. .. .. ... ... ....... . .... .. ...... ......... .. 96 5 AN OUTCOMESPACE CUTTINGPLANE ALGORITHM FOR LINEAR MULTIPLICATIVE PROGRAMMING ......... ....... ... . .. .. ...... 98 5 1. futroduction .. .. .. .. .. .. ..... .. .. .............. ..... . .. ..................... . .. ... ........ ......... 98 5 .2. Theoretical Prerequisite s ..... ..... ................. ............. ... .. . ........... .. .. ... 100 5.3 Out comeSpace, CuttingPlane Algorithm . .. ........ ... . .. ................ . ........ 104 5 .3. 1 Strict Local Optimal Solution Search .............. . ....... ........ .. . ...... 105 5 .3 .2. Cutting PI ane Con s truction ... ... .. . .. .. .. ................. . .... ...... .. ... .... 107 5 .3 .3. Termination Test . ....... .. .. . ........... ......................................... ..... I 09 5 .3.4. Outcome Space CuttingPlane Algorithm ................ ............ ..... 110 5 4. Implementation .. .. .. .... .. .. ....... ..... .. .. ..... ........... ... .. ... ................... .... .. .. 114 5.5. Example .. ......... ..... .. .......... ........ . .. .. .. ........... ................................... .... 119 5 6. Concluding Remarks ... .. ..... . ......... .. .. .. .......... ... .. ..... ............. ..... .. ........ 124 6 SUMMARY AND FUTURE RESEARCH ............................... .. ........... .. ... .. 125 6.1. Introdu ctio n ... .... ... ........ ....... ... . .. ...... .... ......... .... ...... .. .... . .. . .... . . ....... 125 6 .2 Futur e Research on the Heuristic Algorithm ................ .. ................ . ... . 125 6 3. Futur e Re se arch on an Global Solution Algorithms .............. . .... ......... . 127 REIBREN CBS . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 BIOGRAPHICAL SKETCH .. ......... .... .. .. . ................................ ...... ............................ 137 V
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTIPLICATIVE PROGRAMMING: THEORY AND ALGORITHMS By Chairman: Harold P. Benson George Boger December 1999 Major Department: Decision and Information Sciences Multiplicative programming problems are mathematical optimization problems in which the objective function contains a product of several real valued functions defined over a common domain and the feasible decisions are described by a nonempty set. These optimization problems have some important applications in engineering finance, economics, and other fields Multiplicative programming problems, however, are difficult global optimization problems that are known to be NPhard This dissertation has two purposes. The first is to develop and test a heuristic algorithm that finds a good solution, though not necessarily a globally optimal solution, for the linear multiplicative programming problem. The second purpose is to develop a global solution algorithm for the linear multiplicative programming problem that is potentially more efficient than existing algorithms for this problem VI
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To evaluate the effectiveness in practice of the heuristic algorithm, we have written a FORTRAN computer program and used it to solve 260 randomly generated linear multiplicative programming problems of various sizes. Our experimental results show that the computational requirements of the heuristic algorithm are not overly burdensome when compared to the effort required to solve a linear multiplicative programming problem The framework of the outcomespace, cuttingplane algorithm is taken from a pure cutting plane, decision setbased method developed by Horst and Tuy for solving concave minimization problems. By adapting the approach of this method to an outcome space reformulation of the linear multiplicative programming problem, rather than directly applying the method to the original decision set formulation, it is expected that considerable computational savings can potentially be obtained. We also show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem. VII
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CHAPTER 1 lNTRODUCTION 1.1. The Multiplicative Programming Problem Multiplicative programming problems are mathematical optimization problems in which the objective function contains a product of several real valued functions defmed over a common domain and the feasible decisions are describe by a nonempty set. These problems occur is a wide variety of application areas. For example, Konno and Inori ( 1989 ) studied a bond portfolio optimization problem in which the portfolio's performance is measured by a number of indices such as the average coupon rate, the average terminal yield, and the average length to maturity. The goal of the portfolio manager is to improve the performance of the portfolio by purchasing or selling bonds in the marketplace subject to some limiting constraints. The manager must consider multiple incomparable objectives s uch as maximizing the average ter1ninal yield and minimizing the average maturity time. Konno and Inori choose to optimize several objectives simultaneously by multiplying them together since the objectives do not share a common scale. Another example of a multiplicative programming problem, given in Maling, Mueller and Heller ( 1982), is a packaging problem encountered in designing very large scale integrated circuit (VLSI) chips and laying out building floor plans or manufacturing plant facilities. In the problem, the overall rectangular dimensions of the feasible layout 1
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2 plans are constrained rather than fixed. Different layout plans with differing overall rectangular dimensions are obtained according to how the components of a system are arranged within each plan. The objective is to find the arrangement of components that minimizes the overall layout area subject to certain constraints on the area and the perimeter of the layout. Henderson and Quandt (1971, p. 15) also give an application of multiplicative programming problems. Their example is from microeconomics. In their example, a rational consumer wishes to find a combination of two commodities to purchase from which he will derive the highest possible level of satisfaction. Budgetary constraints and the availability of the commodities limit the quantities the consumer may purchase The consumer's level of satisfaction is captured by his utility function, which is assumed to be the product of the quantities of the two commodities. The rational consumer's problem is then formulated as maximizing his utility function subject to the budgetary and commodity availability constraints. The multiplicative programming problem or, more briefly, the multiplicative program, may be formulated mathematically as /J minh(x)= Ilt 1 (x), s t.xe X, }= I where p 2 is an integer, X Rn, and, for each j = 1, 2, ... p, f 1 : X R satisfies f 1 (x) 0 for all x e X. For simplicity we will assume throughout this proposal that the minimum of problem (Px) is achieved at some point x* e X In addition we will assume that p is significantly less than n since this holds for virtually all applications of
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3 multiplicative programming problems. If f j (x) = 0 for some j E {1, 2, ... p} and some x E X, then clearly x is a global optimal solution. This condition can be checked by solving p minimization problems min {J j (x) x E X }, j = l 2, ... p Therefore we may assume without loss of generality that for each j = 1, 2, ... p, f j (x) > 0 holds for all XE X. The objective function h of problem (Px) is generally not a convex function As a result problem ( P x) belong s to a cla ss of nonconvex programming problems called global optimization problems In contrast to convex programming problem s, there may be many local minima for problem (Px) that are not globally optimal. Conventional local optimization method s based on gradients, subgradients, conjugate directions, or the KarushKuhnTucker conditions, for instance, are at best guaranteed only to fmd a local minimum. These methods mu s t then terminate, since there is neither a local criterion for certifying the global optimality of a given solution nor a way to deter1nine how to proceed to a better so lution if the solution is not globally optimal. From the perspective of computational complexity, problem (Px) is a difficult problem that is known to be NPhard even when the objective function is simply h(x) = x 1 x 2 and the feasible region X is a polyhedron (Matsui 1996 ). When in addition to the a ss umptions given previously for problem ( P x ), X is a convex set and for each j = l 2, ... p f j : X R is a concave function we obtain the concave case of problem ( P x ), called the concave multiplicative programming problem. The convex case of problem ( P x ), called the convex multiplicative programming
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4 problem, is obtained when, in addition to the assumptions made previously for problem (Px ), X is a convex set and, for each j = 1, 2, ... p, J i : X R is a convex function. A special linear case of problem (Px), called the linear multiplicative programming problem, is obtained when, in addition to the assumptions make previously for problem (Px ), X is a compact polyhedron and, for each j = 1, 2, ... p, Ji : X R is a linear function (Konno and Kuno 1992). 1.2. Reformulations of the Multiplicative Programming Problem During the 1990's there has been a resurgence of interest in problem (Px ). Encouraged by the rapid advances in high speed computing, researchers began developing and testing new methods for solving global optimization problems that arise in practical applications, including problem (Px ). Included among the global optimization methods used to solve problem (Px) for the special case when p = 2 are various parametric simplex methodbased algorithms (e.g., Konno and Kuno 1992, Konno and Kuno 1995, Konno, Yajima, and Matsui 1991, and Schaible and Sodini 1995), branch and bound procedures (e.g., Kuno 1996 and Muu and Tam 1992), and various other types of algorithms ( e.g., Konno and Kuno 1990, Pardalos 1990, and Tuy and Tam 1992). When p > 2, globally solving problem (Px) has been shown empirically to require considerably more computational effort than when p = 2 (see, e.g., Ryoo and Sahinidis 1996). A smaller number of the algorithms for solving problem (Px) when
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5 p > 2 solve the problem directly without reforrnulating it as an outcomespace problem. Included among these, for instance, is the polyhedral annexation algorithm of Tuy (1991). Most of the algorithms for solving problem (Px) when p > 2, however, solve the problem indirectly by globally solving an outcomespace reformulation of the problem instead. This is because in practical applications p is routinely much smaller than n, often by two or more orders of magnitude. As a result, working in R P is computationally less challenging than working in R n Let y E R P denote the pvector with )th entry equal to y j j = I, 2, ... p. For each j = 1 2, .. p, let y j E R satisfy where y j = +oo is po s sible, and let y E R P denote the vector with )th entry equal to y j )=1,2, .. ,p. Let J(x) denote the vector J(x)=l/i(x),J 2 (x), . ,JP(x)] r where f j : X R, j = 1, 2, . p, are the functions used in defining problem (P x ). Thoai (1991) and later Konno and Kuno (1995) based their outer approximation algorithms for respectively solving the convex and linear cases of problem (Px) on one of the more direct refo11nulations of problem (Px) as an outcomespace problem Their reformulation is given by min (I y j s.t. y E y s j= l where y s = {_y ER P J(x)~ y~ y for somexE X }.
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6 Falk and Paloc sa y (1994) based their branch and bound, image space algorithm for the linear case of problem ( P x) on another outcomespace refo11r1ulation that is closely related to problem ( P r ~ ). Their reforr11ulation is given by min [I yi, s .t. ye Y j = l where Y = { y e R P y = Cx forsomexE X J and C is a ( p x n ) matrix who s e row s are c~, j = 1, 2 ... p. 1.3. Purpose and Organization of the Dissertation This dissertation has two main purpo s es The fust is to develop and test a heuristic algorithm that finds a good solution, though not necessarily a globally optimal solution, for the linear case of problem ( P x ). The s econd purpose is to develop an exact global solution algorithm for the linear case of problem (Px) that is potentially more efficient than existing algorithms for this problem. Since the linear multiplicative programming problem is known to be an NPhard multiextremal global optimization problem, it is inherently more difficult to globally solve than a convex programming problem of the sa me size. In s ome application cases, a solution will adequately meet the requirement s of a user; see, e g ., Konno and Inori (1989). In these cases the use of a heuristic algorithm seems to be appropriate for finding a satisfactory solution. To date however there is no known heuristic algorithm tailored to finding a good solution for the linear multiplicative programming problem In their review of algorithm s for solving problem ( P x ), Konno and Kuno (1995) do not mention
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7 any heuristic algorithms for problem (Px ) and our survey of the literature has revealed none. To develop the heuristic algorithm, we first analyze the concave multiplicative programming problem The analysis yields a new way to write a concave multiplicative programming problem as a concave minimization problem. As a result, a concave multiplicative programming problem can be solved by using any existing concave minimization algorithm without resorting to a reformulation of the problem We also show that some relationships exist between concave multiplicative programming problems and certain multipleobjective mathematical programs. These relationships are exploited to develop the heuristic algorithm for the linear case of problem (Px ). For cases where a linear multiplicative program must be solved for an exact global optimal solution we expect that globally solving the outcomespace refor1r1ulation (P s ) Y instead will result in a significant decrease in the computationa l effort over that required to directly solve the problem. This is because in typical applications of linear multiplicative programs, p is s everal orders of magnitude smaller than n As a result, working in R P should be computationally less challenging than working in R n To globally solve the outcomespace refor111ulation (PY ~ ) of a linear multiplicative program, we develop an outcomespace, pure cutting plane algorithm that works in RP. The framework for the algorithm is taken from a pure cutting plane, decision setbased concave minimization method developed by Horst and Tuy (1993). We show how to adapt this method to solving the reformulation (PY ~ ) of a linear multiplicative program for a global extreme point optimal solution. Once this global solution is found, we can
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8 recover a globally optimal solution for the linear multiplicative program in decision space. As a further computational enhancement, we also show that for purposes of implementation, the mechanics of the outcomespace, cuttingplane algorithm can be applied to the smaller problem (Py) instead of problem (PY ~ ). The organization of the proposal is as follows. In Chapter 2 we present a review of the literature on multiplicative programming problems. In Chapter 3 we analyze the concave multiplicative programming problem, apply the results to develop a heuristic algorithm for the linear multiplicative programming problem, and report test results using the heuristic algorithm on some randomlygenerated problems. In Chapter 4 we analyze the reformulation problem (PY ~ ) and show that, under certain convexity assumptions on Y ~ problem (P Y ~ ) has a global extreme point optimal solution y* E f ~ We then present a procedure that is guaranteed to find a strict local optimal extreme point solution for the reformulation problem (Pr ~ ) of the linear multiplicative program. In Chapter 5 we present an outcomespace cuttingplane algorithm for globally solving a linear multiplicative program. The algorithm employs the strict local optimal search procedure presented in Chapter 4. We also illustrate the algorithm by applying it to the solution of a sample problem Finally, in Chapter 6, we give an overall summary and conclusions, and we discuss directions for further research.
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CHAPTER2 A REVIEW OF THE LITERATURE ON MULTIPLICATIVE PROGRAMMJNG PROBLEMS 2.1. Organization of the Literature Review In this chapter we present a review of the literature on methods proposed for solving multiplicative programming problems. The only known literature review on multiplicative programming problems appears in Konno and Kuno ( 1995 ) In their literature review Konno and Kuno defined multiplicative programming problems as ''a class of minimization problem s containing a product of several convex functions either in its objective function or in its c onstraints They included problems in which the objective function contained the summation of a convex function and the product of convex functions Konno and Kuno ( 1995 ) organized their literature review based on whether the problem data are linear or non] inear and on the number of functions that appear in the objective function They considered solution methods for the following multiplicative programming problems. The first multiplicative programming problem considered by Konno and Kuno is the special case of quadratic programming (LMP2) minf(x) = ((c 1 x)+d 1 )((c 2 ,x)+d 2 ), s.t. XE D where D : = { x E Rn Ax b, x 0} is a nonempty polytope (bounded polyhedron) in 9
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10 which A is an mxn matrix, bE R"', and, for each i = l, 2, c' E R n\ {o} and d; ER. In addition, it is assumed that, for each x E D, ( ci, x) + d; > 0, i = l, 2. The second multiplicative programming problem that they considered is the convex multiplicative programming problem (CMP) min f(x) = Ilt i (x), s.t.xE X, j=I where X Rn is a nonempty, compact, convex set and, for each j = 1, 2, ... p, Ji : Rn R is a convex function that satisfies J i (x) > 0 for all x e X. Konno and Kuno (1995) considered two special cases of problem (CMP): (1) the case where p = 2 and (2) the case where p 2 and the problem data are linear. The second case may be defined as the following extension of problem (LMP2): (LMP) minf(x)= Ii[(c;,x)+d ; ], s.t.xe D, i= l where p 2 is an integer and, for each i = 1, 2, .. p, ( ci, x) + d; > 0 holds for all x e D. Finally, Konno and Kuno (1995) considered three classes of problems related to problem (CMP) In the first class is the following problem: (GCMP) minf(x)=f 0 ( x)+ ft 2 i_ 1 (x)f 2 i (x), s.t.xE X, }=I where, for each j = 0, I, ... 2q, J i : R R is a convex function that satisfies J i (x) > 0 for all xe X. The second class is a special case of (GCMP) in which q = l and the problem data are linear. This class may be defined as the following extension of problem (LMP2) :
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11 (GLMP) minf(x ) =(c 0 ,x)+((c 1 ,x)+d 1 )((c 2 ,x)+d 2 ), s.t.xe D, where c 0 e R 11 and ci, d ;, i = 1, 2, and D are defined as in problem (LMP2). The third class of problems considered by Konno and Kuno (1995) is the minimization of a convex function over a feasible region that includes a product of convex functions in its constraint set. Konno and Kuno's coverage of the literature is not exhaustive. They focused on algorithms that have been demonstrated by computational experiments to be practical for reasonably large problems ( Konno and Kuno 1995, p. 370). Algorithms proposed by Konno, Kuno and their associates have been te s ted on randomly generated problems and the results reported. However, computational results have not been reported by most of the other researchers and therefore their methods were not included in the review Since the publication of the review by Konno and Kuno, two more multiplicative programming problems have been discussed in the literature The first problem adds a convex function to the objective of problem (LMP2) to obtain the problem: (CLMP) minf (x)=g(x )+( (c 1 ,x)+ d 1 )((c 2 ,x)+d 2 ), s.t.xe D, where g : R R is a twice differentiable convex function and ci, d ;, i = 1 2, and D are defined as in problem ( LMP2 ) The second problem adds a convex function to problem (C MP) to obtain the problem: (CC MP ) min f(x)=J 0 (x)+ ]]J i (x), s.t.xE X, j= I where / 0 : R R is a convex function that satisfies / 0 (x) > 0 for all x E X and f j, j = 1, 2, .. p, and X are defined as in problem (CMP).
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12 The emphasis of this review will be on optimization problems in which a product of functions appears in the objective function. Optimization problems with objective functions that are comprised of a summation of a function and the product of functions are also included in the review. Methods proposed for solving these problems may be adapted to solve a problem whose objective function is strictly a product of functions by setting the added function to the null function. The functions that appear in the objective function will be either convex or linear functions since to date these are the only multiplicative programming problems to appear in the literature. In this review we will not consider optimization problems in which a product of functions appears in the constraint set. Like the review of Konno and Kuno (1995), this literature review is organized based on whether the problem data are linear or nonlinear and on the number of functions that appear in the objective function. It is divided into the following four sections. Section 2.2 reviews the methods proposed to solve problems (LMP2), (GLMP), and (CLMP). Section 2.3 reviews the methods to solve problem (LMP) that are extensions of methods for problem (LMP2). Section 2.4 reviews the methods to solve problems (CMP), (GCMP), and (CCMP). Section 2.5 reviews the methods to solve problem (LMP) as a concave minimization problem. The rationale for organizing the literature review in this way is as follows. Historically, the first algorithms for solving multiplicative programming problems were specifically proposed for solving problem (LMP2). Problems (GLMP) and (CLMP) are grouped with problem (LMP2) since they were conceived as extensions of that problem. Several of the algorithms proposed for solving problem (LMP2) can be extended to solve
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13 the problem (LMP), since they do not depend upon having only two functions in the product term of the objective function. Problems (LMP2), (LM P ), (GLMP), and (CLMP) contain linear functions and polyhedral feasible regions. Algorithms for solving these problems are implemented with the aid of the simplex method, which is used to solve linear programming subprob l ems. The problems (CMP), (GCMP), and (CCMP) contain nonlinear data and must rely on other optimization methods to solve nonlinear convex programming problems. The latter three problems are therefore placed in a separate group. Problems (GCMP) and (CCMP) are included in the group with problem (CMP) because only one article addresses each problem, and they were conceived as extensions of problem (CMP). Finally, two articles appeared in the literature that proposed solving problem (LMP) as a concave minimization problem using techniques that the authors had previously developed. Table 2.1 gives a summary of the multiplicative programming problems considered in this literature review along with the assumptions placed on the feasible region and the objective function of each problem. 2.2. Methods to Solve Problems (LMP2), (GLMP), and (CLMP) The methods for so lvin g problem (LMP2), (GLMP), and (CLMP) are further divided into four categories. In the first category are those methods that analyze problem (LMP2) as a special case of quadratic programming. In the second category are algorithms that analyze problem (LMP2) by searching the outcome set. In the third category are the algorithms that solve an easier parametric programming problem rather than directly solving problems (LMP2), (GLMP), and (CLMP). In the last category are
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Table 2.1. Summary of Multiplicative Program Types and Assumptions on Problems Problem Assumptions on the Objective Function Assumptions on the Objective Function Feasible Region LMP2 D is a bounded polyhedron. (c 1 ,x)+d 1 (c 2 ,x)+d 2 ( c;, x) + di > 0, i = 1, 2 for all x E D GLMP D is a bounded polyhedron. (c 0 ,x)+((c 1 ,x)+d 1 )((c 2 ,x)+d 2 ) ( c O x) > 0 and ( c i x) + d ; > 0 i = 1, 2 for all XE D. CLMP D is a bounded polyhedron. g(x )+ (( c 1 x) + d 1 )(( c 2 x) + d 2 ) g: R n R is a twice differentiable convex function and ( ci, x) + d i > 0, i = 1, 2, for all XE D. LMP D is a bounded polyhedron. 11 [(C i X) + d;] (c ; ,x)+d ; >O, i=l,2, ... ,p, for all xE D. i = I ll1 j(x) For each j = 1, 2 ... p, Ji : R 11 R is a CMP X is a compact convex set. convex function that satisfies f i (x) > 0 for all j=I XE X. fo (x) + f f 2 jt (x )f2i (x) For each j = 0, 1, ... p, f i : R n R is a GCMP X i s a compact convex set. con vex function that satisfies f i (x) > 0 for all j=I XE X. fo(x)+ 111 i (x) For each j = 0, 1, ... p, Ji : R 11 R is a CCMP X is a compact convex set. convex function that satisfies f i (x) > 0 for all j=I XE X.
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15 two algorithms th a t s olve problem (LMP2 ) based on the method of polyhedral annexation. 2.2.1. Methods Based on Quadratic Programming Since the objective fun c tion of problem ( LMP2 ) can be expressed as f(x) = ( (c' ,x) + d )( ( c 2 ,x ) + d )= r Qx + r r x + d 1 d 2 where r E R n and Q i s a real s ymmetric n x n matrix, problem (LMP2) i s a special class of quadratic pro g ramming Swarup ( 1966a and 1966b ) was the first researcher to analyze problem (LMP2 ) in thi s way but he did not propo s e any exact solution algorithms. His two articles are included in the literature review for completeness Pardalo s ( 1990 ) also analyzed problem (LMP2 ) in this way and he proposed an exact global solution algorithm Swarup ( 1966a ) showed that if both linear functions ( c ;, x ) + d ;, i = 1, 2 are positive over the feasible regi o n D the objective function f is quasiconcave over D It i s well known that generally for any local minimizer of a quasiconcave function over a polytope, there exi s t s an extreme point local minimizer over the polytope that has the s ame function value Swarup proposed a s implex based method for finding such a local optimal solution. The key to the algorithm is a test that deterrr1ines if entering a given nonbasic variable into the current simplex ba s i s will lower the objective function value. A simplex basis of a local optimal s olution can be reached by beginning at any feasible basis and moving through a s equence of simplex tableaux by pivoting in qualifying nonba s ic variable s until none r e main Once a local optimal solution is found, the
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16 algorithm stops. No inforrnation is available to either certify the global optimality of the solution or to determine how to proceed to an improved solution. In another work, Swarup (1966b) formulated the following parametric linear program by introducing an auxiliary variable and moving one of the linear functions into the constraint set: (MPl) min F(x;~) = ( c', x) + d 1 s.t. xED, Since ( c 2 x) + d 2 appears in the constraint set, dual pricing information is available to determine the value of ( c 1 x) + d 1 as is set to achievable values of ( c 2 x) + d 2 over D. Swarup derived a test that uses this inforrnation to detern1ine when is set to a level that corresponds to a local optimal solution. All local optimal solutions can then theoretically be found by parametrically solving problem (MPl) over all achievable values of~A global optimal solution x of problem (LMP2) can then be found by identifying a global solution (x, ~) of problem (MPl). Pardalos (1990) observed that if c 1 and c 2 are linearly independent, then the Hessian matrix Q of the objective function of problem (LMP2) has one positive eigenvalue and one negative eigenvalue, and the remaining eigenvalues are equal to zero. By applying the spectral decomposition theorem of linear algebra, the objective function can be rewritten in terms of two variables. The problem can then be solved by examining the vertices of an orthogonal projection of the feasible region D into a twodimensional
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17 polytope in the space of the two variables used in the rewritten objective function. Pardalos (1990) proposed an algorithm that enumerates all vertices of the two dimensional polytope until an optimal vertex is found The algorithm may require an exponential number of s teps, but its average computational time complexity is bounded by a polynomial 2.2.2. Methods Based on Searching the Outcome Set The objective function of problem (LMP2) can be expressed as the composite lJI(
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18 feasible region. The approximations are used in a series of optimization problems that are easier to solve than the original problem These optimization problems are sequentially solved until a global optimal solution to the original problem is found. The technique has been very useful in solving global optimization problems in which the feasible region Z is a polytope and the global optimal solution is known to be an extreme point of Z In this f ortn of outer approximation, the algorithm begins by finding a simple polytope P 0 :::> Z with an ea s ily defined inequality representation and an easily calculated set of vertices. A s e rie s of a l g orithmic iterations follows that builds a sequence of decreasing polytopes Po :::> Pi :::> :::> Z in which one polytope is generated in each iteration. In an iteration k of the algorithm the original objective function is evaluated at the extreme points of P k to find an optimal solution v k If v k is an extreme point of Z then v k is a global optimal solution to the original problem. Otherwise a portion of JJ,. \ Z is cut off to fortn P k+I The point v k i s part of the region cut off; i.e. vk is not included in the polytope Pk +I The cut is made by adding a constraint called a cutting plane constraint to the con s traint set that defines P k The cutting plane constraint adds additional vertices to ~ + i that were not pre s ent in P k and the ref ore they must be calculated. Aneja Ag g arwal and Nair ( 1984 ) propo s ed an algorithm that examines the solutions associat e d with the bicriterion programming problem: ( BCP ) VMIN ( y 1 = (c 1 x ) + d 1 y 2 = ( c 2 x ) + d 2 ), s.t xe D.
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19 The intent of problem (BCP) is to simultaneously minimize the two criterion functions y 1 and y 2 Conflicts usually exist between the two criterion functions that prevent a single point of D from simultaneously minimizing both functions. The usual notion of an optimal solution used in single objective linear programming is replaced by the concept of efficient solutions when discussing the solutions of problem (BCP). A solution x is an efficient solution of problem (BCP) if x e D and, whenever for each i = 1, 2, ( c;, x) + d ; ( ci, x) + d ; for some x E D then ( c;, x) + di = ( c;, .x) + d i i = 1, 2. The set of efficient points of D is mapped by
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20 algorithm is then repeated using each of the smaller triangles. The algorithm tertninates when there are no more extreme points of the efficient frontier that need to be searched. In the algorithm of Aneja, Aggarwal, a nd Nair (1984), a new vertex must be calculated for each triangle This is easily done by solving two systems of two equations in the unknowns y 1 and y 2 This special technique however, can not be easily extended to handle cases where p > 2. Falk and Palocsay (1994) also proposed a solution algorithm that searches among the extreme points of Y using a modified outer approximation technique. In the first phase of the algorithm the two linear programs l 1 = min(c 1 x) + d 1 and 1 2 = min(c 2 x) + d 2 x eD x eD are solved for optimal solutions x 1 and x 2 respectively. Two initial vertices y 1 and y 2 of Y are then An initial polytope in outcomespace containing an optimal solution for the problem (YP ) 2 min IJy i ye Y I 1 = such that a 1 y 1 + a 2 y 2 = 1 passes through the point k { i I( l J ) ( 2 2 )J Y = ar~n Y Y1 Y 2 Y1 Y 2 1= 1 ,2 In each iteration of the algorithm, values for a 1 and a 2 are updated and a linear program of the form
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21 (YLP) is solved to remove portions of the initial polytope from the search for an optimal solution for problem (YP) The new vertices generated at each iteration are easily calculated since the isovalue contours of problem (YLP) are linear. The algorithm terminates when the optimal value of problem (YLP) is one. The algorithm proposed by Thoai ( 1991) for solving problem (LMP2) uses an outer approximation technique that begins by enclosing the outcome set Yin a rectangle P 0 In an iteration k of the algorithm, the extreme point (v 1 v 2 ) of the outer approximation that yields the lowest value of the product y 1 y 2 is found. A linear program is then used to deterr1tine if the extreme point (v 1 v 2 ) maps to a feasible point x of D. If not, information is obtained from the linear program to generate a cutting plane constraint that slices off the extreme point (v 1 v 2 ) from the polytope Pk The new vertices generated by the cut are then calculated using a conventional approach (see Horst, Pardalos, and Thoai 1995 or Horst and Tuy 1993). Since the method of deter1nining these new vertices is not dependent on the fact that the dimension of the outcome set is two Thoai' s algorithm can be extended to handle cases where p > 2. In the algorithms of Aneja, Aggarwal, and Nair (1984) and Thoai (1991), the only variations in the linear programs used in successive iterations involve changes in objective function coefficients The authors gain some computational efficiency by restarting the simplex method at the optimal solution of the previous iteration. Only a few simplex pivots are then generally needed to produce a new optimal solution.
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22 2.2.3. Methods Based on Solving a Parametric Master Problem The difficulty in solving problem (LMP2) is caused by the product forrn of the objective function. Konno and Kuno (1992) added a parameter g and formed the following problem that they called the master problem: (MP2) min F(x;~) = ~((c' ,x) + d,)+f ((c 2 x) + d2), s.t. xE D, g ~O. Notice that for a fixed value g' of g, problem (MP2) is a linear programming problem. To solve problem (MP2), Konno and Kuno proposed using a parametric objective function simplex method to the find critical values of g at which new bases become optimal. The values of the objective function F are then evaluated at these bases. A global optimal solution (x g ) of problem (MP2) is found by choosing the basis that minimizes F over these values. Konno and Kuno ( 1992) showed that if (x, g) is an optimal solution of problem (MP2), then x* is a global optimal solution of problem (LMP2). Konno and Kuno tested this algorithm on randomly generated problems (LMP2) with nonnegative problem data that ranged in size from (m, n) = (30, 50) to (220, 200). Their computational experiments showed that the amount of computational time needed to solve problem (LMP2) is not much different from that required to solve linear programs of the same size. In Konno and Kuno ( 1995) the authors slightly simplified the above parametric method by redefining the auxiliary parameter so that convex combinations of the two linear functions are used in the objective function of problem (MP2). This modification
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23 makes it easier to find critical parameter values, since the interval [o, 1] over which the auxiliary parameter ranges is bounded. The rest of the method remained the same. Although Konno and Kuno (1992) did not explicitly say it, their algorithm can be viewed as searching the efficient extreme points of problem (BCP) for one that is a global optimal solution of problem (LMP2). Notice that for a sufficiently small value ~,, an extreme point optimal solution (x', ~') to problem (MP2) coincides with an optimal solution x' of the linear program min { ( c 2 x) + d x E D} Similarly, for a sufficiently large value ... an extreme point optimal solution (x..,, ~*) coincides with an optimal solution x'' of the linear program min {(c 1 ,x)+ d 1 XE D} For any fixed value~> 0, the objective function F( x,~) is a composite objective function for1ned by multiplying the two linear functions by positive values and summing the result. It is well known that any extreme point minimizer of such a composite objective function over the feasible region D is an efficient extreme point of the problem (BCP) (Steuer 1986). The efficient extreme points of problem (BCP) are found by solving linear programs for parameter values between ~, and .... As Aneja, Aggarwal and Nair (1984) have shown, the global solution lies at an efficient extreme point of D in problem (BCP). A disadvantage of the algorithm of Konno and Kuno is that it may require many pivots to solve problem (MP2 ) for all possible parameter values. This will especially be true if there is a great conflict between the two linear functions of the objective function. If for example c 2 = c 1 then every extreme point of D is an efficient extreme point of problem (BCP). Since the size of the set of extreme points of the polytope D grows
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24 exponentially with D, the number of optimal solutions to problem (MP2) over the entire range of parameter values grows exponentially with D and is not bounded by a polynomial. Konno and Kuno in fact observed that the computational time increased as the number of local minima increased. An additional disadvantage of the Konno and Kuno algorithm is that many of the pivots performed will be unnecessary when they are to bases that do not improve on a previously found solution. In another paper, Konno and Kuno (1990) added a convex function to the objective function of problem (LMP2) to obtain the problem (CLMP). With this addition, the objective function may no longer be quasiconcave and therefore, the global minimum may not necessarily be attained at an extreme point of the feasible region D To solve problem (CLMP), Konno and Kuno (1990) proposed an algorithm that solves a parametric master problem which, for a fixed parameter value, is a nonlinear convex programming problem. The algorithm involves solving this master problem a finite number of times, once for each of a finite number of prechosen values for the parameter. A troublesome aspect of the algorithm is that it is difficult to determine the proper parameter values to choose. The authors suggested choosing values for the parameter that are equally spaced in the interval of possible parameter values and solving the resulting master problems to determine a neighborhood containing a globally optimal solution to problem (CLMP). A local search is then done in that neighborhood for a globally optimal solution using the KarushKuhnTucker conditions Care must be taken however, to attempt to define the spacing between the points to be small enough so that a global optimal solution is not missed.
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25 The difficulty that Konno and Kuno ( 1990) encountered in their method in determining parameter values can be eliminated if we assume that the convex function g in the objective function of problem (CLMP) is a linear function. Problem (GLMP) is obtained by making this replacement. Konno, Yajima, and Matsui (1991) considered problem (GLMP), but they assumed that d 1 and d 2 are zero. To solve problem (GLMP), Konno, Yajima, and Matsui formulated the master problem (MP3) min F(x;~) = (c 0 x) + ~( c 2 x), s.t. xe D, Notice that the parameter appears in both the objective function and in a righthand side of a constraint. Konno, Yajima, and Matsui ( 1991) showed that x is a global solution of problem (GLMP) if (x, ~) is an optimal solution of problem (MP3). Schaible and Sodini (1995) used problem (MP3) to show that a global optima] solution of problem (GLMP) lies on an edge of D. Konno, Yajima, and Matsui (1991) proposed a parametric simplex algorithm that includes a righthand side analysis and an objective function analysis to detertnine intervals of parameter values for which bases remain both feasible and optimal. The parametric analysis sweeps through parameter values from ~min= min{(c' ,x)lx e D} to ; max = max~ c 1 x) x e D }. The objective function F is then minimized over each of the intervals.
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26 Konno, Yajima, and Matsui (1991) tested their algorithm on randomly generated problems of up to 350 constraints and 300 variables. They found that the problems can be solved in much the same computational time as that of solving linear programs of equal size. The algorithm of Konno, Yajima, and Matsui (1991) suffers from the same disadvantages as the algorithm of Konno and Kuno ( 1992). In particular, its efficiency depends on the number of pivots performed to solve problem (MP3) for all possible parameter values Also many of the pivots perfor1ned will be unnecessary when they yield bases that do not improve on a previously found solution. Schaible and Sodini (1995) improved the algorithm of Konno, Yajima, and Matsui (1991). From a given simplex tableau for problem (MP3), Schaible and Sodini used parametric analysis to derive a for111ula that calculates the value of the objective function Fas the constraint ( c 1 x) = ~, is set to increasing values of g'. As ~, increases, parametric righthandside analysis calculates new values for the basic variables. Schaible and Sodini then derived some optimality conditions that detect when the parameter ~, is set to a value such that from an optimal solution (x', g') of problem (MP3), one obtains a local minimum x' of problem (GLMP). By applying these optimality conditions, Schaible and Sodini were able to develop a simplexbased algorithm that solves problem (MP3) in a finite number of primal and/or dual simplex iterations. The algorithm proposed by Schaible and Sodini ( 1995) has three advantages over the algorithm of Konno, Yajima, and Matsui (1991): (1) It may terminate before the maximum possible parameter value grnax has been reached. (2) It is more efficient in that
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27 it may skip over local optimal solutions that do not improve the objective function value. (3) It can be used even when the feasible region is unbounded, and it can detect when problem (GLMP) is unbounded from below. Muu and Tam (1992) also considered problem (CLMP), but in their work, the feasible region Dis relaxed to a compact convex set. They seem to be the only researchers to have considered this generalization of problem (CLMP). The authors however tested their algorithm using a polytope for the feasible region. Muu and Tam (1992) formulated the parametric master problem (MP3') s.t. xeD, ( c 1 x) + d 1 = ~, 0. They proposed a branch and bound algorithm to solve problem (MP3'). Branch and bound is a technique commonly used by algorithms in global optimization. Branching refers to the successive partitioning of the feasible region and bounding refers to the computation of lower and upper bounds on the global optimum over the partitions. Partitions of the feasible region that produce a lower bound on the objective function that exceeds the best upper bound found so far by the algorithm are eliminated from further consideration. Such partitions are said to be fa thorned. A branch and bound algorithm terminates when all of the partitions have been fathomed. In the algorithm of Muu and Tam (1992), partitions of the feasible region are constructed by restricting the value of ( c 1 x) + d 1 to values within an interval. The algorithm begins by finding an interval I O := [g 1 g 2 ] of achievable values of ( c 1 x) + d 1
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28 by solving the two convex programs ~ 1 := min { ( c 1 x) + d 1 x E D} and ~ 2 := max {(c 1 x) + d 1 x ED}. Optimal solutions u 0 and v 0 are then obtained for the two convex programs A lower bound /3 (I O ) over the interval / 0 of the objective function F of problem (MP3') is found by selecting /3 (I O ):= min {/3 (~ 1 ), /3 (~ 2 )} An upper bound a 0 on F is obtained by selecting a 0 := min {t(u 0 ), J(v 0 )}. The interval / 0 is next bisected and the procedure repeated using the two subintervals. A subinterval that produces a lower bound that exceeds the current upper bound is eliminated from further consideration; i.e. that subinterval is considered to be fathomed. The procedure continues bisecting intervals / k to generating a sequence of solutions {xk = 1 that converge to a limit point x that is a global optimal solution. Computational experiments on problems up to (m, n) = (30, 200) showed that the algorithm is very efficient when both vectors c and d are positive. 2.2.4. Methods Based on Polyhedral Annexation A limitation of conventional optimization methods is that they can become trapped at a local minimum, or even a stationary point, if they are applied to a global optimization problem, e.g. see the algorithms proposed by Swamp (1966a, 1966b). The central problem of a global optimization method then is to overcome this limitation by providing a certification test for global optimality, and if a point is not globally optimal, deterrnining how to move to a better solution. Tuy (1991) called this the subproblem of
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29 ''transcending the incumbent'' where the incumbent is the best feasible solution found so far by an algorithm. Let f be the objective function for problem (LMP2), and let x be a vertex of D that represents th e incumbent s olution for this problem. Then, from Tuy ( 1991 ) to transcend the incumbent, one must find a point in x e D such that f (x) < f (x) or else e s tablish that no s uch point exi s ts, i.e. that x is a global optimal solution for problem (LMP2). Let G : = { x e n J(x) '?: J (x)}, where n is a convex set containing D The problem of transcending the incumbent can then be restated as the following problem ( GCP ) Check if D c G and if not, find a point x e D \ G Problem (GCP ) i s known as the Geometric Complementary Problem. Tuy ( 1990 ) deve l oped the method of polyhedral annexation to solve problem ( GCP). In polyhedral annexation a sequence of polytopes Pi_ c P 2 c c P k c is built by adding a vert e x to the polytope P kI of the previous iteration in such a way that a vertex of D i s annexed into the new polytope P k The sequence Pi n D P 2 n D ... forrns an expanding inner approximation of D. When a polytope P h D is found all of the extreme point s of D have been searched and the algorithm terminates. A ss ociated with the sequence of p o lytopes Pi c P 2 c c P c is the sequence of their polars Pi :) P 2 *:) :) ~ :) where a polar E of a convex set E in R 11 is defined as E :={ye R n (y, x) 1 for all x e E } A dual correspondence exists between the facets of a polytope P k and th e vertice s of its polar P "* The subproblem of determining the
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30 inequality representation of P k, after a new vertex has been added can then be solved by solving the easier problem of computing the vertices of P; The termination condition D has the corresponding condition ~ k D For a more detailed description of polyhedral annexation, see the chapters on inner approximation in Horst, Pardalos, and Thoai (1995 ) or in Horst and Tuy (1993). Tuy and Tam (1992) proposed two algorithms that are derived using the polyhedral annexation method with a dualization and dimension reduction technique developed by Tuy ( 1991) Dualization refers to solving the original problem by solving the dual problem of generating a sequence of polars until a polar Ph* k D is found. The key to the dimension reduction technique is the introduction of a cone into problem (GCP). Tuy and Tam ( 1992) assumed that c 1 and c 2 are linearly independent vectors and then forn1ed the c one K := {x E R n ( c ; x) 0, i = 1, 2 }. Cone K is of interest since if .x e D is an incumbent solution, then for any .x E (.x + K) f (x) f (x) In other words, cone K identifies points in R 11 that can do no better than the incumbent solution x. Computational effort might be saved using cone K since a part of the feasible region D can be eliminated from further consideration and the search narrowed to the remaining portion of D The first algorithm proposed by Tuy and Tam (1992) solves problem (LMP2) by solving problem ( GCP) through the dualization process of generating a sequence of polars until a polar P, ; k D is found. Tuy and Tam (1992) showed that the polar K* of cone K is explicitly given a s K = {y E R" y = t 1 c 1 t 2 c 2 for some t 1 0 t 2 0 }. Any
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31 vertex y in a polar ~ lies in the polar cone K*, and the multipliers t 1 and t 2 used to express y are unique, since c 1 and c 2 are linearly independent vectors. Polar cone K* is used to solve the dual problem by building a collapsing sequence of polars Pi* ::, P; ::, ::, P; ::, with each polar being an improved approximation of v. The search is conducted in the twodimensional space generated by c 1 and c 2 rather than in the original ndimensional space Solving the linear program (LP(t)) max {fi(c 1 ,x)r 2 (c 2 x)xe DJ, where ti and t 2 are the multipliers used to express some vertex x = Fie' t 2 c 2 of P h* tests for the termination condition Ph* D*. The second algorithm proposed by Tuy and Tam ( 1992) is motivated by the observation that for a fixed value of t = (t,, t 2 ), problem (LP(t)) is equivalent to the linear program (LP(a)) max { (Cl a (c 2 CI X )) X E D} where a= t 2 / (t 2 +t 1 )e [o, 1]. The first algorithm thus reduces to solving a sequence of linear programs (LP( a)) for different values of the parameter a. The second algorithm proposed by Tuy and Tam (1992) is to parametrically solve problem (LP(a)) for all of the critical values of a at which new bases become optimal. The objective function! of problem (LMP2) is evaluated at each basis and a global optimal solution chosen from those bases. The second algorithm of Tuy and Tam (1992) is essentially the same parametric problem (MP2) used by Konno and Kuno (1992).
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32 Tuy and Tam (1992) ran computational experiments using both the first polyhedral annexation algorithm and the second parametric algorithm. Their results showed that for solving problem (LMP2), the parametric algorithm performed better than the polyhedral annexation algorithm. The polyhedral annexation algorithm is not as efficient because more simplex pivots were required than for the parametric algorithm. Tuy and Tam ( 1992) proposed an improved variant of the polyhedral annexation algorithm that reduces the number of pivots and the number of objective function evaluations. The authors observed that the improved algorithm may potentially be more useful for a problem with an objective function that is difficult to evaluate. The computational experiments run using the parametric algorithm on problems of up to (m, n) = (30, 200) and positive problem data were in line with the results reported in Konno and Kuno (1992). 2.3. Extensions of Algorithms for Problem (LMP2) to Solve Problem (LMP) when p~3 The polyhedral annexation method of Tuy and Tam (1992) and the outcomespace algorithms of Thoai (1991) and Falk and Palocsay (1994) can be extended to the more general problem (LMP) where p 3. Although the algorithms remain unchanged, the subproblem of determining the new vertices becomes more difficult as the number of function terms in the objective function increases. 2.4. Methods to Solve Problems (CMP), (GCMP), and (CCMP) Relatively little work has been done in designing exact global solution algorithms that address problems (CMP), (GCMP), and (CCMP). The algorithms that have been
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.. 33 proposed fall into two categories: (1) methods based on solving a refor1nulated problem and (2) a method based on outer approximation. 2.4.1. Methods Based on Solving a Reformulated Problem Konno and Kuno (1992 ) introduced problem (CMP) where p = 2 and for111ulated a master problem by introducing a parameter into the original problem to separate the two functions of the objective function into a summation. This technique of embedding the original problem into a problem in a higher dimensional space is similar to the one used by the authors in the s ame paper to solve problem (LMP2). At the time, Konno and Kuno were not able to give an algorithm for solving the master problem. In Kuno and Konno ( 1991) the authors proposed a branch and bound algorithm along with an underestimation function to solve it. Computational results for problems of up to (m, n) = (200, 180) indicated that the algorithm is efficient when the objective function is the product of a linear function and a quadratic function and the feasible region is a polytope. Kuno Yajima, and Konno (1993) extended the paramaterization technique of Kuno and Konno ( 1991) for problem (CMP) to handle cases where p 2. They showed that a global optimal sol ution to problem (CMP) can be obtained by solving the equivalent problem (MP4 ) mm ~E S where E = ~ER P ]] ~ i 1, 0 For a fixed f EE, let x*(f) denote an optimal j= l solution of min G(x;f)= f ~J i (x). Let h: E R be defined by h(~):= G(x (~); ~) for xeX I 1 =
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34 any c; E 3. Solving problem (MP4) then reduces to solving the problem in RP given by (MP4') mi_!l h(c;). ~ e .::. Kuno, Yajima, and Konno (1993) showed that his a concave function over 3 and therefore a global optimal solution of problem (MP4') exists on the boundary of 3 They proposed an outer approximation method for solving problem (MP4') and tested their algorithm against two subclasses of problem (CMP): (1) problem (LMP) and (2) problems similar those tested in Kuno and Konno (1991) in which the objective function is the product of a linear and a quadratic function and the constraints are linear inequalities. Computational experiments showed that the total computational time is dominated by that needed for solving the convex minimization master problems for each parameter value. The results also showed that the number of cuts and vertices generated increases rapidly as p increased from 2 to 5. The authors asserted that this was due to inefficiencies in computing new vertices, especially when p exceeds 5. However, if p is held constant, these numbers increased very slowly as the number of constraints and variables increased. The authors concluded that their algorithm is reasonably efficient when p is less than 4. Jaumard, Meyer, and Tuy (1997) added a convex function to the objective function of problem (CMP) to for1n problem (CCMP). The authors showed that problem (CCMP) can be reduced to a quasiconcave minimization problem in RP that is a generalization of problem (MP4') used by Kuno, Yajima, and Konno (1993). In the special case where / 0 = 0 in problem (CCMP), the reduced quasiconcave minimization problem in Jaumard, Meyer and Tuy (1997) can be shown to be equivalent to the one
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35 used by Kuno, Yajima and Konno (1993). Jaumard, Meyer, and Tuy (1997) find a global solution of problem (CCMP) by finding an optimal solution to the quasiconcave minimization problem in R P using a conical branch and bound method. They ran computational experiments using their algorithm on test problems similar to those used by Kuno, Yajima and Konno (1993) and Thoai (1991). The authors report that their results are very sensitive to the magnitude of p and not as sensitive to the size (m, n ) of the constraint matrix. Sniedovich and Findlay ( 1995) analyzed problem (CMP) from the perspective of cprogramming but did not give a complete algorithm for solving it. Cprogramming is a technique developed by Sniedovich (1984) for solving an optimization problem of the form ( CP ) q := min l/f(cp(x )), xeX where X is some nonempty set
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36 For problem (CMP), the objective function can be expressed as the composite 1/f( 0 }. Since problem (CMP) satisfies the requirements of cprogramming, it can be solved by solving the parametric problem (MPS') q(~):= rninf ~;/;(x), ~ES, ~ X i= I where S is any subset of R P such that V 1/f(
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37 objective function of problem (GCMP) The master problem is a convex minimization problem in the space R 11 + 2 q and is solved using an outer approximation algorithm. Computational experiments conducted using a polyhedron for the feasible region showed that for q = 1, this algorithm required less than half the computational time required by the branch and bound with underestimation function algorithm proposed in Konno and Kuno (1992) to solve problem (CMP). Tuy (1992 ) gave problem (CMP) as an example of an optimization problem that can be formulated as a Geometric Complementary Problem and solved it using a parametric programming problem. The parametric programming problem is a convex minimization problem in which a positive parameter vector is used to build a composite objective function from the convex functions in the objective function of problem (CMP). A complete algorithm that includes solving the parametric program was not given. 2.4.2. A Method Based on Outer Approximation Thoai ( 1991) extended the algorithm based on the outer approximation technique that he proposed for solving problem (LMP2) to address the solution of problem (CMP) when p = 2. The main idea is to build a sequence of decreasing polytopes P 0 :::) Pi :::) :::) X of the convex feasible region X and a sequence of decreasing ~ ~ polytopes S 0 :::) S 1 :::) :::) Y of the outcome set Y, where Problem (CMP) is then solved by applying a modified version of the algorithm for problem (LMP2). In any iteration k, up to two cuts are introduced, one for Pk and one for S k to obtain tighter approximating sets.
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38 Since the algorithm does not depend on tl1e actual value of p it can be extended to handle cases where p 3 2.5. Methods to Solve Problem (LMP) as a Concave Minimization Problem Konno and Kuno (1992) showed that the objective function of problem (LMP) is not a convex function over the feasible set D Therefore, problem (LMP) is not a convex programming problem. However, since the natural logarithm function In is a strictly increasing concave function on (0, 00 ), it is easy to show that the function defined for all x e D is a concave function. In addition, the optimal solution set of the concave minimization problem (CMIN) min F(x), s.t.xe D, is identical to the optimal solution set of problem (LMP). Therefore, any concave minimization method may be applied to problem (LMP) if the objective function is replaced by its logarithmic equivalent. Using the above transformation modification, Tuy (1991) showed that problem (LMP) could be solved in a reduced dimension space using polyhedral annexation and the dualization and dimension reduction technique. The algorithm presented in Tuy and Tam ( 1992) is essentially an improvement of the one in Tuy ( 1991 ). Ryoo and Sahinidis (1996) also converted problem (LMP) into the problem (CMIN). To solve problem (CMIN), they employed a branch and bound algorithm that incorporates the use of valid inequalities to accelerate convergence. Branch and bound
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39 algorithms may slowly converge to an optimal solution when the gap between the initial upper and lower bounds is large. A valid inequality is a inequality constraint that does not exclude any solution that yields an objective function value lower than the current best upper bound. By introducing valid inequalities into the constraint set, inferior parts of the feasible region may be removed from further consideration without eliminating possible global optimal solutions. A second use of valid inequalities is to reduce the range of values that the variables in the problem can assume. Ryoo and Sahinidis referred to these two uses of valid inequalities as range reduction mechanisms. The performance of the bounding procedure in the branch and bound algorithm is improved by using these range reduction mechanisms, since smallersized partitions of the feasible region are used and the variables are restricted to reduced ranges of values. Ryoo and Sahinidis implemented the branch and bound algorithm along with the range reduction mechanisms in a computer program called BARON (BranchAndReduce Optimization Navigator). To more easily calculate lower bounds on the objective function F of problem (CMIN) over a partition of the feasible region, the authors replaced F by a linear underestimating function. Lower bounds were then calculated by solving linear programs. The authors tested randomlygenerated problems in sizes from (m, n) = (50, 50) to (200, 200), with p ranging from 2 to 5. They reported that only a small fraction of the total CPU time is consumed in the range reduction mechanisms and that there seemed to be a loworder polynomial relationship between the CPU time and the value of p
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CHAPTER3 CONCA VE~MULTIPLICATNE PROGRAMMING PROBLEMS: ANALYSIS AND AN EFFICJENT POINT SEARCH HEURISTIC FOR THE LINEAR CASE 3.1. Introduction An important, but little re s earched area that deserves more attention, is the development of heuri s tic algorithms for finding a good solution for multiplicative programming problem s In s ome applications a good, though not nece ss arily globally optimal solution may adequately meet the requirements of a u s er (Konno and Inori 1989 ) In these ca s es s ince multiplicative programming problems are known to be NP hard the expenditure of computational effort required to globally solve them may not be needed. Thi s chapt e r has two purpose s. The first is to pre s ent an analysis of problem ( P x) when problem ( P x) i s a conca v e multiplicative programming problem. The second purpose is to propo s e a heuristic algorithm de s igned for the case where problem ( P x) is a linear multiplicative programming problem. The analy s i s of the concave multiplicative programming problem i s presented in Section 3 2 This a naly s is show s a new way to write a concave multiplicative programming problem as a concave minimization problem and s ome theoretical consequence s of this. It also s hows some relationships between concave multiplicative programs and certain multipleobjective mathematical programs. In Section 3 3 by using some of the re s ult s of Section 3.2, we present and explain the workings of an efficient40
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41 point search heuristic algorithm that we have developed for the linear multiplicative programming problem. Section 3.4 reports and analyzes some statistics summarizing the computational re s ults that we obtained by coding the heuristic algorithm and applying it to 260 randomlygenerated linear multiplicative programs. In Section 3.4 we also report the results of applying the heuristic algorithm to a multiplicative programming problem fo1med from a decision situation using real data. In Section 3.5, we discuss the major results of thi s chapter. 3.2. Analysis Assume in problem ( P x) that X i s a convex set and that, for each j = 1 2, ... p, f j : X R is a concave function; i e., assume that problem (Px) is a concave multiplicative programming problem Consider the function g : X R defined for each XE X by g(x)=log g(x) Then, it is a simple matter to s how that g : X R is a concave function and that the optimal solution s et to the concave minimization problem ming(x), s t xe X, ( 3.1) is identical to the optimal solution set of problem ( Px ) Thus, any concave multiplicative programming problem of the for1n of problem ( Px ), if rewritten in the fortn (3 1), can be solved by applying any appropriate generalpurpose concave minimization algorithm to (3.1). For discussions and reviews of concave minimization algorithms, s ee, for instance, Benson ( 1995 ), B e n s on ( 1996 ) Horst and Tuy ( 1993), and Pardalos and Rosen (1987 ).
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42 It is interesting and useful in both practice and theory to observe that, in addition to (3.1), there is at least one other way to rewrite a concave multiplicative programming problem as a concave minimization problem. To show how this can be accomplished, we will first prove the following preliminary result Lemma 3.2.1. Let a E R P satisfy a > 0, and consider the nonlinear programming problem v=min(a,A), s.t.AEA, (3.2) where A= A E R P [[ A j 1, A~ 0 Then, vis finite and problem (3.2) has at least one J = I optimal solution. Proof. Notice that, if AE A, then A> 0 and (a,1) > 0. Therefore, v > 0. This, combined with the fact that A :t 0, implies that v is finite. Now, suppose that, for each j = 1, 2, ... there exists a vector 1 1 E A such that where { e j 4 1 is a strictly decreasing sequence of positive real numbers such that l ime 1 = 0. Then the sequence { A j l i is either bounded or unbounded. J oo J Case 1: { A j h =1 is bounded. Then, for some bounded set A k A, A j E A for each j = 1, 2, ... Therefore by passing to an appropriate subsequence { A1 }je, of { Aj 1 1 if t A necessary, we can guarantee that A = lim A1 exists. Further1r1ore, since A1 E A k A for }e l each j E I, and A is a closed set, A belongs to A By assumption, (3.3)
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43 for each j E J. By talcing the limits over j E J on both sides of (3.3), we conclude that ( a, I)~ v Since IE A, this implies that I is an optimal solution to (3.2). Case 2: { Al h = t is unbounded. Then, for some subsequence { ). / 1eJ of {Al};: 1 and for some k E { 1, 2, ... p }, l im A{ = + 00 For each j E J, since ).,i E A, ).,i > 0. 00 Combined with the fact that a > 0 implies that for each j E J, ( 3.4) By assumption, for each j E J, ( 3.5) From (3.4) and (3.5), we obtain ( 3.6) for each j E J. By taking the limits over j E J on both sides of (3.6), we conclude that + oo = v, which is a contradiction. Therefore, this case cannot hold, and the proof is complete. Using Lemma 3.2.1, we may now establish the following theorem. Theorem 3.2.1. Assume in problem (Px) that X is a convex set and that Ji : X R, j = 1, 2 .. p, are concave functions. Let g: X R be defined for each x e X by p g(x) = p IT 1 1 (x) 1 /p j=I Then g : X R is a concave function.
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44 Proof. C o nsider the function h : X R defined for each x E X by h( x ) = min f A ifJ (x ) s .t. A EA ( 3.7 ) j=I where A is as de f ined in Lemma 3.2.1. From Lemma 3 2.1 since J i is s trictly positive on X for each j = 1 2, ... p it follows that the minimum in ( 3.7) exist s and i s finite for each x E X. If f o r each A E A we define a function h ). : X R by h ;t (x)= f i 1 J i (x) j = l then for each x E X h(x) may also be written as h(x) = min h ;t ( x ). ..leA ( 3.8 ) Notice that for each A E A h i : X R is a concave function. From this and ( 3.8 ), we conclude that h: X R i s al s o a concave function (Rockafellar 1970 ). To complete the proof we will show that, for each x E X h(x) = g ( x ) Toward this end, fix x E X and let A( x )e X denote an optimal solution to problem (3 7 ) From the KarushKuhn Tucker necessary condition s for this problem ( Bazaraa Sherali, and Shetty 1993 ), s ince A(x) > 0, it follows that there exists a nonnegative constant 0 (x) such that p J j ( x )e(x) rri k ( x ) A j (x)=O, j=l,2 .. p. ( 3.9) k = l Since A(x )e X i s an optimal solution to problem ( 3.7 ) it is easy to see that
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45 Together with (3.9), this implies that A1 (x )J 1 (x) = 0(x ), j = l, 2, ... p. From (3.10), it follows that A 1 (x)=0(x)/ J 1 (x), j = I, 2, ... ,p. By substitution in this implies that 1/p 0(x) = 11 f 1 (x) j=l From equations (3.10) and (3.11), we see that 1/ p f A 1 (x)J 1 (x )= p [IJ 1 (x) j= I }=I (3.10) (3.11) (3.12) Since x e X and A(x )e X is an optimal solution to (3.7), the lefthandside of equation (3.12) coincides with h(x). By definition of g, the righthandside of equation (3.12) equals g(x ), so that the proof is complete. Theorem 3.2.1 can also be proven by using a composite function approach and showing several preliminary results ( Avriel, Diewert, Schaible, and Zang 1987). We offer the proof here, because it is more direct and because we will use it below to help derive a corollary of interest.
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46 Notice from Theorem 3.2.1 that, when problem (Px) is a concave multiplicative program, the optimal solution set of problem (Px) is identical to the optimal solution set of the concave minimization problem min g(x), s.t. xe X, where g: X R is defined for each x E X by g(x)= p[g(x)] 1 1 P. (3.13) In practice, this implies that any concave multiplicative program (Px ), if rewritten in the fotm (3.13), can be solved by app lying any suitable concave minimization algorithm to (3.13). Notice also that problem (3.13) is a simpler reformulation of problem (Px) for the concave case than the typical refor1r1ulation used in the literature to solve problem (Px) in the convex case (see e.g., Konno and Kuno 1992, Kuno and Konno 1991, Thoai 1991, and Kuno, Yajima, and Konno 1993). Theorem 3.2.1 also has some interesting theoretical implications concerning the product of functions. For instance, for any finite set of concave functions fj, j = 1, 2, ... p, each defined on a common nonempty convex domain X R n and each strictly positive on this domain, it is known that the function g : X R defined by their product is not necessarily concave, convex, or quasiconvex on X (Kuno, Yajima and Konno 1993 and Avriel, Diewert, Schaible and Zang 1988). However, from Theorem 3.2 .1 the function f : X R given by J(x) = [g(x )] 1 1 P for each x e X is a concave function on X
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47 In addition, Theorem 3.2.1 implies the following result concerning the product of a set of concave functions. Corollary 3.2.1. Let X and f j j = 1, 2, ... p, be defined as in Theorem 3.2.1, and suppose that g : X R is defined for each x E X by g(x) = fI f 1 (x). j=I Then g: X R is a quasiconcave function. Proof. Choose a e R and let La = {x E X g (x) a}. If a~ 0, La = X is a convex set. If a> 0, then from Theorem 3.2.1 and Rockafellar (1970), the set [ 13 = {xe X p[g(x)] 1 1 P /3} is a convex set, where /3 = pa 1 IP. Since [ 13 =L a this implies that La is a convex set. Therefore, we have shown that for any a E R, La is a convex set. This is equivalent to showing that g : X R is a quasiconcave function (Bazaraa, Sherali, and Shetty 1993), so that the proof is complete. It fol]ows from Corollai y 3.2.1 that any concave multiplicative programming problem (Px) is a problem involving the minimization of a quasiconcave function over a convex set. Many of the most popular algorithms for minimizing a concave function over a convex set are equally suitable for minimizing quasiconcave functions over convex sets (Horst and Tuy 1993 and Benson 1995). As a result, we see that any concave
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48 multiplicative program (Px) can be solved by applying any number of suitable concave minimization algorithms directly to problem (Px ). In particular, no refo1rnulations of problem (Px) are needed to apply these algorithms. Remark 3.2.1. Corollary 3.2.1 has been previously shown to hold for the special case where p = 2, X is a nonempty, compact polyhedron, and / 1 and / 2 are linear functions (see, e.g., Konno and Kuno 1992). The next corollary of Theorem 3.2.1 concerns the minimization problem (3.7) used in the proof of the theorem. Possible uses for this corollary may include the construction of methods for finding local optimal solutions to concave multiplicative programs, although we will not investigate this here. Corollary 3.2.2. Let X and J i j = 1, 2, ... p, be defined as in Theorem 3.2.1, and let A be defined as in Lemma 3 .2 .1. Then, A is a convex set and, for each x E X, the unique optimal sol ution l(x) to problem (3.7) is given by 1 / p \(x)= Ii ti(x) fk(x), k=l,2, .. ,p. j = I Proof. Notice that A may be rewritten according to the relation 1 / p A = ;t E int R : p Ii Ai p (3.14) j = 1 where It is easy to see that, for each j = 1, 2, ... p, h i : int R : R defined for each ;t E int R : by
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is a concave function on int R : that satisfies Therefore, by Theorem 3 .2.1, the function m : int R: R defined for each A E int R: by 1 / p is a concave function. This implies that is a convex set (Rockafellar 1970). By (3.14), this proves that A is a convex set. Now, fix xe X, and let A(x)e A denote an optimal solution to problem (3.7). From the proof of Theorem 3.2.1, this implies that, for each k = 1, 2, ... p, where e(x) is given by (3.11 ), so that the corollary is proven. In addition to its relationships to concave minimization, a concave multiplicative program also has some interesting ties to multipleobjective mathematical programming. In the remainder of this section, we will show some of the theoretical relationships between concave multiplicative programs and certain multipleobjective mathematical programs. In the next section, some practical benefits of those relationships will be demonstrated. Let J(x) denote the vector Vi (x ), f 2 (x ), ... J P (x )]r,
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50 where J 1 : X R, j = I, 2, ... p, are the functions used in defining problem (Px ). Then, the components of the vector f (x) are generally conflicting, in the sense that the infima over X of J 1 (x ), j = 1 2, ... p, are generally not simultaneously achieved at the same point in X. As a result, inherent trade offs in the achievable values of the components of f (x) over x e X are present. To account for these tradeoffs, and to seek what decision makers call a most preferred solution in situations where the goal is to attempt to simultaneously minimize J 1 (x ), j = 1, 2, ... p, over X, one of the most popular approaches is to consider the associated multipleobjective mathematical program VMIN J(x ), s.t. xe X. (3.15) In particular, in typical situations, a most preferred solution in X will exist that is also an efficient solution for (3.15), where an efficient solution is defined as follows. Definition 3.2.1 A point x 0 E Rn is called an efficient solution for (3.15) when x 0 E X and, whenever f (x)~ f (x 0 ) for some xE X, then J(x) = J(x 0 ). An efficient solution is also called a nondominated or Paretooptimal solution. By generating or searching the set XE of the efficient solutions for (3 .15), decision makers are able to observe the inherent tradeoffs among the objective functions J 1 j = 1, 2, ... p, that are available over X and are often able to choose from XE a most preferred solution. For further discussions on multipleobjective mathematical programming and its applications, the reader may consult, for instance, Cohon (1978), Evans (1984), Luc
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51 (1989 ) Sawaragi Nakayama a nd Tanino (1985 ), Stadler (1979 ), Steuer ( 1986 ) Yu ( 1985), Zeleny ( 1982 ) and references therein. The fir s t relationship between multiplicative programming and multipleobjective mathematical programming is given in the following re s ult. The proof of this result is an elementary exercise Proposition 3 2 1 Any optim a l s olution to problem ( P x) must belong to the efficient set X E of the multipleobjective mathematical programming problem (3.15 ). Notice that Pr o position 3.2.1 holds for arbitrary multiplicative programming problems ( P x ) The next result however is re s tricted to certain types of concave multiplicative program s. Proposition 3 2 2 A ss ume in problem ( P x ) that X is a compact, convex s et and that J i : X R j = 1 2, . p ar e concave functions Then there exists an optimal solution to problem ( P x) which i s an extreme point of X. Proof From Theorem 3 2 1 problem ( P x) can be solved by finding an optimal solution to the concav e minimization problem ( 3 13 ), where g: X R i s the concave function defined by p g(x) = p IJJ i (x) 1 / p j= I for each xe X Since X i s a nonempty compact convex set, from Horst and Tuy ( 1993) problem ( 3.1 3) ha s an optimal s olution that is an extreme point of X. The s e two observations together prove th e de s ired re s ult
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52 Taken together, Proposition 3.2.1 and 3.2.2 imply that any concave multiplicative programming problem with a compact feasible region has at least one optimal solution that is an efficient extreme point solution to the multipleobjective mathematical programming problem (3.15). Special cases of this observations have been alluded to in the literature (see, e.g., Aneja, Aggarwal and Nair 1984 and Sniedovich and Findlay, 1995). In the next section, we put this observation to practical use. 3.3. Efficient Point Search Heuristic Assume in this section that, in problem (Px ), X = {x E R 11 Ax b} is a compact polyhedron, where A is an m x n matrix and b E Rn', and that for each j=l,2, ... ,p, J j (x)=(c j, x), where c j ER 11 foreach j=l,2, ... ,p. Then problem (Px) is a linear multiplicative programming problem or, more briefly, a linear multiplicative program (Konno and Kuno 1992). We have designed and tested a heuristic algorithm for this problem, based in part on some of the results in the previous section. In this section, we will formally state this heuristic algorithm and explain its workings. The multipleobjective program (3.15) associated with a linear multiplicative problem may be written as VMIN Cx, s.t. Ax~ b, (3.16) where C is the p x n matrix whosejth row equals (c j f, j = 1, 2, ... p. Problem (3.16) is a multipleobjective linear programming problem (Steuer 1986 and Yu 1985). Let X ex denote the set of extreme point s of
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53 X = { x E Rn Ax b }. Then, by Proposition 3.2.1 and 3.2.2, an optimal solution to the linear multiplicative programming problem can be found in the set of efficient extreme points of problem (3.16). The set X E ,e x is finite, and various procedures have been developed for generating it in its entirety (see, e.g. Steuer 1986, Yu 1985 and Steuer 1983). It follows that in theory at least, a global optimal solution to a linear multiplicative problem can be found by completely enumerating the set X E,ex of efficient extreme points of the associated multipleobjective linear programming problem (3.16) and, from this set, choosing the point(s) with the smallest value of (see, e.g. Sniedovich and Findlay 1995). Unfortunately, as we shall see Jater, in practice the exponential growth in the size of X E,ex as a function of problem size (Steuer 1986) renders this approach impractical for many cases. The approach of the heuristic algorithm is to efficiently search a dispersed, carefully chosen sample of candidate points from X E,ex in order to find an attractive solution to the linear multiplicative programming problem. To describe and explain the workings of the heuristic, we must first present some theoretical background from the theory of multipleobjective linear programming. Let
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54 W = {w E RP ( e, w) M, w e }, where e E R P is a vector with each entry equal to 1.0, and Mis a positive real number. For sufficiently large M, from Philip (1972) it is known that a point x 0 belongs to the efficient set XE of (3.16) if and only if x 0 is an optimal solution to the weightedsum problem (3.17) for some w = w 0 E W We will assume henceforth that Mis chosen to be large enough to guarantee that this property bolds. It is also well known that the efficient set XE for (3.16) is given by where, for each w E W, Xi v denotes the optimal solution set of the linear program (3 .17) (Steuer 1986 and Yu 1985) Since the optimal solution set to (3.17) for any w E W is a face of it follows that the efficient set XE for (3.16) is equal to the union of the faces Xi v, we W, of X. Although X E is a connected set (Yu 1985), it is generally nonconvex. The heuristic algorithm will individually identify efficient faces Xw, we W, of X, and find an approximatelyoptimal extreme point solution to the problem p min IJ (ci,x), s.t.xE X 1 v (3.18) j=l for each efficient face X w that it finds.
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55 Let Y = { y e R P y = C x, for somexe X }, y >= { y eR P y ~ y, forsome y eY} To aid in its search, the heuri s tic algorithm will solve the linear program s t. Cx y Ax~b ( 3.19a ) (3 19b) (3.19c ) for various value s of y E y > and w E W The heuristic relies in part upon the properties of problem (3.19 ) given in the next three results The first two results follow easily from Benson ( 1978 ) Theorem 3.3.1. Suppose that x 0 E R n and let y 0 = Cx 0 Then, x 0 is an efficient solution for (3.16 ) if and only if with y = y 0 x 0 is an optimal solution to (3.19 ) for every weW Theorem 3.3 2 If y E y > and w E W, then (3 19 ) bas at least one optimal solution and any optimal s olution for (3 19 ) is an efficient solution for (3 16). Theorem 3.3.3. Suppose in (3 19 ) that w = w 0 E W and that y = y 0 = Cx 0 where x 0 is an efficient solution for ( 3.16 ) Let (u 0 r z 0 r) denote any optimal solution to linear programming dual of ( 3.19 ) where u 0 repre s ents the dual variables corresponding to the constraints C x y 0 of ( 3 19 ). Let w 0 = u 0 + w 0 and let v 0 = (w 0 J Cx 0 Then x 0 belongs to the efficient face X :cc0 of X and X :cc0 can be represented as IV W
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56 X w = {x E X (w 0 r Cx = Vo Proof. To prove the theorem, we will show that, with w = w 0 x 0 is an optimal solution to problem (3.17). Suppose in (3.19) that w= w 0 e W and that y = y 0 = Cx 0 where x 0 is an efficient solution for (3.16) given in the theorem. The dual linear program to (3.19) is then given by max(y 0 ,u)(b,z), C T AT er 0 s.t. u z = w u,z 0. From Theorem 3.3.1, x 0 is an optimal solution to (3.19) when w= w 0 and y = y 0 By the duality theory of linear programming (Murty 1983), since (u 0 r, zor) is an optimal solution to the linear programming dual of (3.19) when w = w 0 and y = y 0 this implies that (wo)r Cxo =(yo,uo)(b,zo). By rearranging this equation and using the definitions of y 0 and w 0 we obtain (w 0 f Cx 0 = (b, z 0 ). With w = w 0 the dual linear program to (3.17) may be written as max(b, z), A T cT0 s.t. z = w z ~O. (3.20) (3.21a) (3.21b) (3.21c) Let z denote an arbitrary feasible solution to problem (3.21). From the definitions of u 0
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57 and w 0 this implie s that (u 0 r, zr ) is a feasible solution to the dual linear program of (3 19 ) Since (u 0 T, z 0 r ) is an optimal solution to the latter problem, it follow s that (yo, u o) ( b, zo) ~ ( y o u o) ( b, z ) or equivalently ( b z 0 ) ( b z). Notice that since (u 0 r zor ) i s an optimal s olution to the dual linear program to ( 3 19 ) z 0 i s a feasible s oluti o n to ( 3.21 ). By the choice of z, the preceding two s tatements imply that z 0 is an optimal s olution to ( 3.21) Since x 0 is an efficient solution for (3 16 ) with w = w 0 x 0 i s a fea s ible s olution for ( 3.17 ). From ( 3.20) and the duality theory of linear programming ( Murty 1983 ) s ince z 0 is an optimal s olution to (3.21) this implies that, with w = w 0 x 0 i s an optimal s olution to ( 3 17 ), and the proof is complete Notice in Theorem 3 3 3 that, for any t > 0, X = = X 0 This implies that in tw w Theorem 3 3 3 when w 0 W there exist s a t E (0 1) such that tw 0 E W and X 0 = X 0 Thu s, in Theorem 3.3 3 when w 0 W X 0 has an alternate representation tl V lV lV X =o for which w 0 E W For s implicity we may and will assume without loss of I V generality that in Theorem 3. 3 3 w 0 E W. To generat e various p o ints y E Y ~ for u s e in problem (3.19) the heurist i c algorithm will rel y upon the two concepts defined in the next two definitions (s ee e.g., Zeleny 1982 ).
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58 Definition 3.3.1. The point y 1 E R P is called the ideal point of Y when, for each j = 1, 2, ... p, y equals the minimum value of y j over Y. Definition 3.3.2. The point yAI E R P is called the antiideal point of Y when, for each j = 1, 2 ... p y; 1 equals the maximum value of y j over Y. Notice that y~ and y; 1 generally do not belong to Y. The algorithm uses these two points as anchor point s in an initialization procedure whose goal is, in part to generate a dispersed sample of points from Y ~ The heuristic algorithm may be stated as follows. Algorithm 3 3.1 Efficient Point Search Heuristic Algorithm Initialization Phase. See Step s 1 through 5 below. Step 1. Find the ideal and antiideal points y 1 and yAI of Y. Step 2. Find an optimal solution l(x ) r ,a J E R n+ t to the linear program max a, a~O Step 3. Choose a positive integer S and, for each i = 1, 2, .. S, let Step 4. Choose a positive integer N such that 1 N M p + 1, let w 0 = e ER P, and, for each j = 1, 2, ... p define w 1 E R P by
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. w != I Step 5. Set UB = + 00 i = 0 and j = 0. 59 1, if i i= j, N, if i = j. Efficient Point Search Phase. See Steps 1 through 6 below. Step 1. Set y = y; and w = w i, and find any optimal solution x ii to linear program (3.19). Step 2. Set y = Cxij and w = w i in (3.19), and compute any optimal solution l(u ij f, ( zii f j to the dual linear program to (3.19), where u ii denotes the optimal dual variables corresponding to the constraints Cx y of (3.19). Step 3 Let w ij = u ii + w i. If w ij is a positive multiple of w i'i' for some i' i and j' j such that (i' j') i= (i, j ), then go to Step 6. Otherwise, continue. Step 4. Let v i} = (w ij f Cx ij. For each h = 1, 2, .. n, calculate a h according to the formula II ( ij) r k] a h = L c 'x LC h k= I t1:k (3.22) .. and find any basic optimal solution x' 1 to the linear program min ( a ,x), (3.23a) (3.23b) Ax~b (3 23c) Step 5. If TI ( c k ,xii)~ UB go to Step 6. Otherwise set x = x ij, and UB = n (ck ,x), k=I k=I and go to Step 6.
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60 Step 6. Set j = j + 1 If j?. p go to Step 1 Otherwise, set i = i + 1 and j = 0. If i S go to Step 1. Otherwi s e, Stop : x E X E.ex is the recommended solution to the linear multiplicative programming problem. In the initialization pha s e of the algorithm s amples of points from Y ~ and from Ware generated. To g enerate the sample of points from Y ~ Step 2 of thi s phase determines the point y* between yA 1 and y 1 such that, of all line segments with endpoints yAJ and y that lie in Y ~ and for which y lies on the line segment connecting y AJ and y 1 the line s egment L connecting yAJ and y has maximum norm. The sample { y; i = 1 2, .. S} of points from Y ~ i s then generated in Step 3 of this phase by partitioning L into S line segments of equal length, where S is a positive integer chosen by the user In Step 4, a sample of p + 1 all integer vectors from W is generated, where for p of the s e vectors the value N of one of the components is chosen by the user from the set { I, 2, ... M p + I} Each iteration of the efficient point search phase of the heuristic executes two key operations. First, it identifies an efficient face X (J of X Second, unless this face has \\I .. been previously identified during an earlier execution of this phase, with w = w u in problem (3.18 ) by u s ing a firstorder linear approximation to the objective function of this problem, it finds an extreme point xif of X in this efficient face that i s an approximate optimal solution to ( 3.18 ) Steps 1 thr o ugh 3 of the efficient point s earch phase of the algorithm identify an efficient face of X In Step 1 with y = y; E y > and w = w j E W, the linear program
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61 (3.19) is solved for any optimal solution x lj By Theorem 3.3.2, this optimal solution .. must exist and is an efficient solution for (3.16). In Steps 2 and 3, with y = Cx 1 and w = w 1 in (3.19 ), the dual linear program to (3.19) is solved to yield the vector uil E R P . . and the weighting vector w lj = u 1 + w 1 is computed. From Theorem 3.3.3, the face X w; 1 corresponding to this weighting vector is an efficient face for (3 .16) and contains x ii. Furthermore, from the same theorem, this face can be written as (3.24) where v ii = (w ii f Cx iJ. Step 3 checks whether or not X wu has been identified during a previous execution of this phase of the algorithm. If so, the algorithm proceeds to Step 6 to prepare for another possible iteration of the efficient point search phase of the heuristic. Otherwise control shifts to Steps 45. In Steps 45 of the efficient point search phase, problem (3.18) is approximately solved using a new efficient face X w as the feasible region. In particular, in Step 4, (3.22 ) is first used to construct the nonconstant portion of a firstorder Taylor series linear approximation ( a,x ) of the objective function of problem (3.18) at x = x iJ E X wu Next, using the representation (3.24 ) of the efficient face X u an extreme point minimizer x ii w of (a, x ) over X \V ij is found by solving the linear program (3.23). Notice that x ii E X E,l! x .. (see Rockafellar 1970 ). In Step 5 the value achieved by x lj in the objective function of the linear multiplicative problem is compared to the smallest value UB found thus far for .. this objective function by the search. If x 1 achieves a smaller objective function value
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62 . than UB, xi) becomes the new incumbent solution x and UB is reduced in value accordingly. Notice that the perforrnance of the heuristic algorithm depends in part upon the number, locations and dimensions of the efficient faces (3 24) that are searched via problem (3.23). This, in tum, is partially dependent upon the sizes of the parameters S and N chosen by the user. The goa l is to search as many points of X E ex as possible by generating a variety of distinct efficient faces (3.24) of large dimensions that are dispersed widely throughout XE. Notice that, since each efficient face identified by the heuristic is given in the form (3.24) and searched by solving linear program (3.23), the individual points in X E e .x that are searched by the algorithm are searched implicitly rather than explicitly, i.e., they do not need to be explicitly enumerated. 3.4. Computational Results The heuristic a lgorithm described in Section 3.3 has the following attractive characteristics: (a) it can be implemented using only linear programming methods; (b) it generally implicitly searches many efficient extreme points of (3.16) at once by optimizing over entire efficient faces of (3.16), rather than by explicitly examining individual efficient extreme points of (3.16); ( c) it allows the user to manipulate the nature and extent of the efficient face search through the choices for the input parameters S and N; (d) it finds efficient faces of (3.16) by attempting to globally sample from a variety of regions of the efficient set.
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63 To evaluate the effectiveness in practice of the heuristic algorithm and its features, we have written a VS FORTRAN computer code for the algorithm and used it to solve 260 linear multiplicative programming problems of various sizes. To execute the code on these 260 problem s we used an IBM ES/9000 model 831 mainframe computer. As a further illustration of the effectiveness in practice of the heuristic algorithm we solved a multipleobject linear programming problem in forest management that was derived from a real decision situation using real data To implement Step 3 of the initialization phase of the algorithm, we chose to set S = 4, so that a sample of five points lying between yAJ and y 1 in Y ~ is always generated in this step. We used a value of N = 9 in Step 4 of the initialization phase to help generate the s ample of p + l points from W. To solve the linear programming problems called for by the heuristic, the computer code use s the simplex method procedures given in the subroutines of the Optimization Subroutine Library (International Business Machines 1990). These subroutines employ anticycling rules to handle degeneracy as needed. Therefore, they are especially appropriate for solving instances of problem (3.23), since these problems always contain degenerate extreme points Let and suppose that k is a positive integer. To generate the 260 test problems, we used the following random procedure First for each j = 1 2, ... p, we generated the elements of the vector ci E R n by randomly drawing elements from the set {1, 2, .. 10} Next, we
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64 generated a nonempty, compact polyhedral feasible region X k int R ;. This region can be written as X J R" P > 1 < < 1 2 } 1:XE x=q, =x i =q,J, ... ,n, where P is a k x n matrix, q E R k, and CJ E R. To accomplish this, first the elements of P were generated by randomly choosing elements from the set { 1, 2, ... 10}. Next, for each i = 1, 2, ... k, the formula II qi= LP i j= I was used to calculate q ; and, finally, CJ was chosen according to the rule q = max {q ; ji = 1, 2, .. k }. Each test problem was constructed to belong to one of four categories, where a category is defmed by the number p of linear functions used in the objective function Ii ( c 1 x) of the test problem. The values p = 2, 3, 4, 5 were chosen to define these J = l categories. We chose these categories in this way because empirical evidence seems to indicate that the complexity of these problems is more sensitive to the magnitude of p than to the magnitudes of k or n (Kuna, Y ajima and Konno 1993). Within each category, the test problems were classified into subcategories of 10 problems, each defined by the values of the ordered pair (k, n). To help evaluate the attractiveness of the solutions found by the heuristic algorithm, we found a global optimal solution for each test problem by completely enumerating all of the efficient extreme points of the associated multipleobjective linear
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65 program (3.16). To accomplish this, we use the ADBASE computer code developed by Steuer (1983) Some stati s tic s summarizing the results of these computations are presented in Tables 3.13 4. In each table, each row gives average statistics for a subcategory (k, n) of 10 problems a measure of the worst case perfor111ance of the heuristic, and the number of problems in a category for which a global optimal solution was found. The first statistic is the average number of efficient extreme point s found by ADBASE in solving the problem s by complete enumeration. In some sense, the magnitudes of the s e numbers correspond to the average relative difficulties, by subcategory of each group of 10 linear multiplicative programs in a subcategory The s econd statistic is the average efficiency rating r given by r = 1 [( z H Z mi n )/( zmax Z min )], where z H i s the objective function value returned by the heuristic and where z min and Zmax are the global minimum and maximum values of the objective function of the test problem over the corre s ponding s et of efficient extreme points of (3.16). Thus 0 r 1 and the closer r i s to 1.0 the more attractive the value z H returned by the heuristic is relative to the actual g lobal minimum value zmin The third statistic given for each subcategory in the s e tables i s the average CPU time ( seconds) that the heuristic needed to solve a problem in the s ubcategory. The fourth statistic shows the lowest efficiency rating calculated for a problem in the subcategory It gives a measure of the wor s t case performance of the heuri s tic algorithm when applied to the 10 problems in a subcategory.
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66 Table 3 1 Computational Results : p = 2. Subcategory Avg No. Avg. Eff. Avg. Solutions Lowest Eff No. Exact k n Eff Points Rating r Time ( sec ) Rating r Solutions 25 20 28.8 1 000 0 227 1.000 10 25 30 2 8.8 1 000 0 241 1 000 10 30 40 4 7 9 1 000 0 389 1.000 10 40 30 2 8 2 1.000 0 328 1.000 10 40 50 47.0 0 999 0 504 0 996 8 50 40 3 5 1 0 999 0 453 0 999 9 50 60 29 2 1.000 0 556 1 000 10 60 70 62 3 1.000 1 070 1.000 10 The fifth statistic i s th e number of pr o blem s in a category for which the h e uristic algorithm found a global optimal solution. These four tables show that the solution s returned by the heuristic algorithm give, on the avera g e, quite accurate estimate s of the actual global minimum values for the 260 linear multiplicative te s t problems generated. This i s indicated by the fact that average efficiency rating s by s ubcategory always were a t least 0 920 and in approximately 96 % of the subcategories e x ceeded 0 950 It i s noteworthy that, for these problem s, these ratings r by s ubcategory do not s eem to decline s ignificantly as p, k, and n increase in Table 3.2. Computational Result s : p = 3. Subcategory A v g No. Eff. Avg Eff. Avg. Solutions Lowest Eff. No Exact k n E xt. Point s Rating r Time ( sec .) Rating r Solutions 25 20 33 0 6 0 985 0.321 0.951 4 25 3 0 896.8 0 960 0.469 0 708 5 30 40 873 3 0.987 0.543 0 884 7 40 30 949.3 0.993 0 609 0 968 6 40 50 2 0 7 3 7 0 920 0 967 0 806 4 50 40 1484 9 0 993 0 908 0 961 7 50 60 2846 3 0 995 1 298 0 978 6 60 70 5867.5 0.969 2.495 0.799 2
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67 Table 3.3. Computational Results: p = 4. Subcategory Avg. No. Eff. Avg Eff. Avg. Solutions Lowest Eff. No. Exact k n Ext Points Rating r Time (sec.) Rating r Solutions 25 20 2789.5 0 998 0 426 0.993 4 25 30 7245.9 0.992 0.598 0.945 5 30 40 23656 0.986 1 019 0.947 1 40 30 19034 0.978 0.998 0.923 2 40 50 50889 0.969 1.539 0.918 0 50 40 59443 0.969 1.587 0.843 2 50 50 83780 0.981 1.901 0.890 3 value. In addition, wjth the exception of one subcategory, a global optimal solution was found for at least one problem in each subcategory. The average solution times by subcategories shown in the four tables indicate that, for these test problems, the computational effort required by the heuristic was rather small. In fact these average times were always less than 2.50 seconds. In comparison to exact algorithms that have been used in test situations to globally solve linear multiplicative problems, these times are generally either at least as small or much smaller (see, e.g., Kuno Yajima, Konno 1993 and Ryoo and Sahinidjs 1996). Furthermore, in contrast to solution times for exact algorithms, these average solution times seem much less sensitive to increases in p n, k or to increases in the average number of efficient Table 3.4. Computational Results: p = 5. Subcategory Avg. No. Eff Avg. Eff. Avg. Solutions Lowest Eff. No. Exact k n Ext. Points Rating r Time t'sec.) Rating r Solutions 10 20 1331.4 0.993 0.353 0.941 5 20 10 527 1 0 .9 98 0.294 0.993 2 25 30 57115 0.995 0.962 0.992 2
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68 extreme points that exist in the corresponding problems (3.16); see Kuno, Yajima, Konno (1993) and Ryoo and Sahinidis, (1996). Finally, it is worth noting that we were able to apply the heuristic to much larger problems than those reported in Tables 3.13.4. However, the number of efficient extreme points in the associated multipleobjective linear programming problems (3.16) for these cases always exceeded 200,000. Since the ADBASE code cannot be used to find all of the efficient extreme points for such problems, we were unable to completely enumerate the sets of efficient extreme points to find zmin and r values for these problems. Thus, we are as yet not able to draw conclusions concerning the accuracy of the heuristic for any problems larger than those reported in Tables 3.13.4. To further illustrate the effectiveness in practice of the heuristic algorithm, we solved a real application problem in forest management that was studied in Steuer and Schuler (1978) as a multipleobjective linear programming problem. The problem involves the allocation of land and budget monies in a way that seeks to maximize objectives in timber production, hunting and cattle grazing in the Swan Creek subunit of the Mark Twain National Forest Steuer and Schuler (1978) provide actual data used to for1nulate their multipleobjective linear programming problem. The problem contains 31 decision variables, 5 linear objective functions, and 13 constraints. Our multiplicative programming problem was formed from this problem by multiplying the 5 linear objective functions together to form a single objective function. The heuristic was then used to search for an approximate solution that maximizes this single objective function subject to the constraints of the forest management multipleobjective linear programming problem.
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69 To help evaluate the attractivenes s of the solution found by the heuristic algorithm we found a global optimal solution by enumerating the 83 efficient extreme points of the as s ociated forest management multipleobjective linear program using the ADBASE computer code An efficiency rating of r = 0.999 was calculated using the slightly modified equation r = 1 [( zmax Z H )/( z max Z min )] since this multiplicative programming problem is a maximization problem rather than a minimization problem This efficiency rating indicates that the heuristic algorithm returned an attractive value zH relative to the actual global maximum value Zmax 3.5. Discussion The result s of this chapter imply that there are at least two ways to rewrite a concave multiplicative programming problem as a concave minimization problem. It f ol]ow s that concave minimization theory and methods can be used in these ways to analyze and s olve concave multiplicative programs. The results also imply that a concave multiplicative programming problem can be analyzed and solved directly without any reformulation, as a qua s iconcave minimization problem over a convex set Furthermore the analysis in the chapter implies that any concave multiplicative programming problem (P x ) with a compact feasible region has at least one optimal solution that is an efficient extreme point solution of the a s sociated multipleobjective mathematical programming problem ( 3.15 ). Therefore the opportunity exists for devising s olution methods for such problems (P x) that search am o ng the efficient extreme points of the associated multiple objective problem s ( 3 15 ) The c hapter proposes a heuristic algorithm that takes this
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70 approach for solving linear multiplicative programs. From the computational results presented for this heuristic algorithm, we conclude that its features and perfor1nance offer significant potential for conveniently finding very attractive solutions with relatively little computational effort to the various applications using linear multiplicative programming encountered in practice Thus, the theoretical and algorithmic results presented in this chapter offer some potential new avenues for more effectively analyzing and solving multiplicative programming problems of various types
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CHAPTER4 A GENERAL MULTIPLICATNE PROGRAMMING PROBLEM IN OUTCOME SPACE 4.1. Introduction Recall from Chapter 1 that the multiplicative programming problem is given by v x =min Ilt 1 (x),s.t xe X, J= l where p 2 is an integer, X is a nonempty set in Rn, and, for each j = 1, 2, ... p, J 1 : X R satisfies J 1 (x) > 0 for all x e X. For simplicity, we assume that the minimum v x in problem (Px) is achieved. For any x e R ", let J(x) denote the pvector withjth entry equal to J 1 (x), j = 1, 2, ... p. Let ye RP denote the pvector withjth entry equal to y 1 j = 1, 2, .. p. For each j = 1 2 . p, let S, 1 e R satisfy where y 1 = + 00 is possible, and ]et ye RP denote the vector withjth entry equal to y 1 j = 1, 2, ... p. Although various outcomespace reformulations of problem (Px) have been proposed for solution purposes, one of the most common reformulations is given by the problem v Y = min g(y), s.t. ye Y ~ 71
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72 where Y ~ = ~ E R P J(x)~y~ y for somexE X ], ( 4.1) and where for e ac h y E Y ~ g : Y ~ R is defined by ( 4.2 ) For example problem ( P Y ~ ) i s essentially the reformulation of problem ( P x ) used in the algorithm s of Ben s on ( 1998c ), Falk and Palocsay ( 1994 ) and Thoai (1991 ). Notice that since X is nonempty Y ~ i s a nonempty s et. By con s tructing appropriate global s olution algorithms for problem ( P Y ~ ), this problem provides us with the opportunity to solve problem ( P x) by working in the outcomes pace R P of the problem, rather than in the decision s pace R 11 which i s generally much larger than R P. In order to globally s olve problem ( P Y ~ ), it is important to understand the properties of the set Y ~ defined by ( 4.1 ) of the function g defined by ( 4.2 ), and of problem ( PY ~ ) itself. This chapter undertake s a mathematical analy s is of the outcomespace reformulation ( P y s ) of problem ( P x ). The analysi s is organized accordin g to whether or not the outcome s pace problem s atisfies conditions for the general ca se, the convex case, or the polyhedral ca s e. For the g eneral ca s e we show for instance that globally s olving either problem ( P x) or problem ( P Y ~ ) e ss entially also globally solves the other problem, and that for any f e a s ible point y for problem ( P y s ), either g ( y ) < g( y ) for some y E Y ~ or y satisfies a c ondition that i s nece ss ary but not sufficient, for it to be a local optimal s olution for probl e m ( P Y ~ ). F o r the convex and polyhedral cases we s how s tronger
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73 results. For example we show for the convex case that any global optimal solution for problem (PY ~ ) must lie on the boundary of Y ~ that the objective function g in problem (P s ) is strictly pseudoconcave on Y ~ and, when Y ~ is closed and contains at least one Y extreme point, that problem (P Y~ ) has an extreme point global optimal solution. The analysis of the general case of problem (PY ~ ) is given in Section 4.2. Section 4.3 provides analytical results for both the convex and polyhedral cases of problem (PY ~ ). 4.2. Results for the General Case of Problem (Py f ) Notice under the assumptions made in Section 4.1 for problem (Px) that Y ~ is a nonempty subset of R : := { z E R P z > 0 }. When y s satisfies this condition, we obtain what we will call the general case of problem (PY ~ ). It is important to establish that by solving the general case outcomespace for1r1ulation (Py f ) of problem (Px ), a global optimal solution for problem (Px) can be recovered. The following result, by showing that problems (Px) and (Pr ~ ) are equivalent in a certain sense, immediately establishes this fact. Theorem 4.2.1. (a) If x is a global optimal solution for problem (Px ), then y = J(x) is a global optimal solution for problem (Pr ~ ). Further1nore, v >' = v x (b) Problem (P Y~ ) has at least one global optimal solution Further1nore, if y* is a global optimal solution for problem (Pr ~ ), then any x* E X such that J(x )~ y is a global optimal solution for problem (Px ).
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74 Proof. (a) Let x be a global optimal solution for problem (Px ) and set y = J(x ). From (4.1) and (4.2) this implies that y* E Y ~ and that Therefore, v Y ~v x If g(y)< v x were to hold for some yE Y~, then, from (4.1) and (4.2), there would exist an x E X such that 0< I11 j (x)~g(y)< v x j= I which contradicts the definition of v x Therefore, g(y )~ v x for all y E Y ~ This implies th > s. < ( ) d y ~ C ll th ( ) d at v Y = v x mce v Y = v x g y an y E 1t 10 ows at v Y vx g y an y 1s a global optimal solution for problem (PY ~ ). (b) By assumption, we may choose a global optimal solution for problem (Px ). From part (a), this implies that problem (Pr ~ ) has at least one global optimal solution. Suppose that y is a global optimal solution for problem (Pr ~ ). Since y E f f (4.1) implies that we may choose an arbitrary x E X such that J(x )~ y Then, from ( 4.2), since O < f (x ), Since x E X and y is a global optimal solution for problem (Pr ~ ), this implies that V x 11 f j (x )~V y (4.3) j= I
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75 From part (a ) v " = v x By (4 3 ), this implies that Ii f j (x )= v x Since x E X it follows }= 1 that x is a global optimal solution for problem (Px ). Suppose in the general case of problem ( Pr ~ ) that a point y E f ~ has been generated For algorithmic purposes, it may be valuable to have a tool for finding an alternate point y E f ~ that sati s fies g(y) < g( y ), if such a point exists. The next result gives an idea for potentially helping to create such a tool. To prove this re s ult we need the following lemma. This lemma will also be u s eful in proving several other results later in this chapter. Lemma 4.2 1. A ss ume that y E f ~. Then for any y E Y ~ (1 / p ) ( V g(y) y) = g(y)(1 / p )f (y j /y j ), J= I and g ( y )(l / p )f (y l /yj ) ~ g (y)[g(y )/ g(y)]l /p }= 1 with equality holding in the latter relationship iff, for some constant M > 0 y 1 = M y 1 j=l,2 ... ,p Proof Choo se an arbitrary point y E Y < Suppose that yE Y ~ Then by ( 4 2 ) since Y ~ R : g ( y ) > 0. By definition of g, (1 / p) ( Vg( y ) y )= (l / p)f ITYk J }= I k~j
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76 (4.4) Since (1/ p )'?:.0, (y i /y 1 )> 0 for each j = 1, 2, . p, and p(l/ p) = 1, the arithmeticgeometric mean inequality (Duffin, Peterson, and 2.ener 1967) implies that j= l with equality holding iff for some constant M > 0, y i = M yi for each j = 1, 2, ... p. Together with (4.4), since g(y)'?:_O, this implies the desired results. Theorem 4.2.2. Assume that y E y s If 1. o > in~ (1/ p) t (y j / y j ) JE Y j =I (4 5) then g(y )< g(y) for some yE y s In particular, if y achieves the infimum in (4.5), then g(y)< g(y). Proof Suppose that y E y < If ( 4.5) holds, then for some y E Y ~ 1. o > ( 1/ p) t (y j / y j ) (4.6) j= l Since g(y) > 0, this implies that g(y) > g(y)(l/ P )t (Y1 /Yi). (4.7) }= 1 From Lemma 4.2.1, since y E Y ~ we know that g(y )(1/ p )t (yi /yi )~ g(y)[g(y)/ g(y)]' I P_ (4.8) j= I Since g(y) > 0, together (4.7) and (4.8) imply that
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77 1.0 > [g(y)/ g(y)]l/p. Because g (y) > 0, this implies that g (y) < g (y). The ref ore, g (y) < g (y) for some y E ys. Since, for any point y that achieves the infimum in ( 4.5), ( 4.6) is also satisfied, the argument above also implies that if y achieves the infimum in (4.5), then g(y)< g(y). Notice that when y E Y ~ the infimum in ( 4 5) is either less than 1.0 or equal to 1.0. From Theorem 4.2.2, when this infimum is less than 1.0, a point y in y s such that g(y) < g(y) exists. In particular, in this case y is not a global optimal solution for problem (PY ~ ). The next result covers the case when the infimum in (4.5) equals 1.0. Theorem 4.2.3. Assume that y E Y ~ If (4.9) then y is an optimal solution to vd =min(Vg(y),yy), ye Y ~ (4.10) Proof From ( 4.9), since y E Y ~ the infimum in ( 4.9) is achieved at y = y. By Lemma 4 2.1, since g(y) is a positive constant, tbjs implies that y also mjnimizes (1/ p )(V g(y), y) over y s Since (1/ p) is a positive constant and (V g(y), y) is a constant, it is easy to see that this implies that y is an optimal solution to ( 4.10) and
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78 A point y E y s is a local optimal solution for problem (PY ~ ) when there exists an > 0 such that for each yE Y ~ for which llyy ~, g(y )~ g(y). From Theorem 4.2.3, when y E y and ( 4.9 ) hold s, then for any y E y s if there is a o > 0 such that d := (y y ) sati s fies y + Ad E Y ~ for all A such that O < A~ o, the directional derivative of g at y in the direction d will be nonnegative i.e ( Vg(y),d)~O. From Bazaraa, Sherali and Shetty ( 1993), thi s is a necessary but not sufficient condition for y to be a local (or global ) optimal s olution for problem (PY ~ ). 4.3. Results for Convex and Polyhedral Cases of Problem (PY ~ ) When y s in addition to being a nonempty subset of R:, is a convex set, then we obtain what we will call the convex case of problem (Pr ~ ). Similarly, when y s in addition to being a nonempty subset of R : is a polyhedron, then we obtain what we will call the polyhedral ca s e of problem ( Pr ~ ) Each of these types of outcomespace versions of problem ( P x ) arise s from a broad class of decision space problems, as s hown by the next result. Theorem 4 3 1 When X is a convex set and, for each j = 1, 2, ... p, f j i s a convex function on X we obtain the convex case of problem ( PY ~ ). When X i s a polyhedron and, for each j = I 2 .. p f j is linear on R ", we obtain the polyhedral case of problem ( P y s ). Proof A ss ume in addition to the assumptions made in Section 4.1 on X and on f j j = 1, 2 .. p that X i s a convex set and that, for each j = 1 2 ... p f j is a
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79 convex function on X. We will show that y < is a convex set. Choose any y 1 y 2 E Y~. From (4.1), since y 1 y 2 E y s, we may choose x 1 x 2 E X such that fj (x 1 )~ y 1 and J j (x 2 )~y 2 }=1,2, ... p. Suppose that le Rand O~l~l. Then,since A~O and (1l)~O, for each }=1,2, ... ,p, l 1 (x 1 )+ (1l )Jj (x 2 )~;., y 1 + (1l )y 2 By the convexity of f j j = 1, 2, ... p, on the convex set X, if we set x = Ax 1 + (1A )x 2 then XE X, and, for each j = 1, 2 . p, ( 4.11) (4.12) ( 4.13) From(4 ll)(4.13 ), J(x)~ly 1 +(Il)y 2 where .xE X .Since y 1 ,y 2 E y s, y; ~Y holds for each i = 1, 2. As a result, since A, (1A)~ 0, A y 1 + {1A )y 2 y The conditions for A y 1 + (1A )y 2 to belong to Y ~ are thus satisfied. By the choices of y 1 y 2 and A, this implies that Y is a convex set. Now suppose, in addition to the assumptions made in Section 4. I on X and on fj, j = 1, 2 ... p that X is a polyhedron and that, for each j = 1, 2, ... p, fj is a linear function on R n We will show that Y ~ is a polyhedron By definition, since X is a polyhedron, there exists a finite number q of linear functions g 1 j = 1, 2, ... q, on Rn, and real numbers b j, j = 1, 2, .. q, such that X = {xE R n g j (x)~b j, j = 1, 2, ... ,q}.
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80 Let Z R n+p be defined as th e se t of all s olutions ( x, y ) to the system of linear inequalities ( 4.14 ) ( 4.16 ) given by )=1,2, ... ,p, ( 4.14 ) j = 1, 2, ... p, ( 4 15 ) gj(x ) < b = J' j = 1 2, ... q. ( 4.16 ) Then by definition Z i s a polyhedron in R "+P Let A be the p x (n + p) matrix whose first n columns each equal O E R P and whose last p columns together form the p x p identity matrix. Then from (4 1 ) and the definition of Z, Y = AZ From Rockafellar (1970, Theorem 19.3 ), Y ~ i s a polyhedron in R P. In convex cases of problem (PY ~ ) ( and thus, in polyhedral cases as well ) certain locations within y s for s eeking global optimal solutions can be specified. For instance we have the following re s ult Theorem 4.3.2. Suppose that problem ( P r ~ ) satisfies the conditions for the convex case. Then: (a) Any g lobal optimal so lution for problem (Py s ) belongs to the boundary of r s. (b) If y s is closed and contains at lea s t one extreme point, then there exists at least one global optimal solution for problem ( P y s ) that is an extreme point of y s Proof. A ss ume that y s in addition to being a nonempty subset of R :, is a convex set, i .e., that we have th e convex case for problem (Pr ~ ) Then from Theorem 4 .2. 1, problem ( P Y ~ ) has at lea s t one global optimal solution.
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81 (a) To show this part of the theorem let y denote an arbitrary global optimal solution to problem (Pr f ). Suppose that y is not on the boundary of Y $. By the choice of y and since Y ~ i s a convex set, y s has a nonempty interior. Therefore y must belong to the interior of Y ~ From ( 4.1), thi s implies that for some xE X, f(x)< y must hold. By assumption, since x EX, J(x)> 0. Therefore, if we set y = f(x), it follows that y E Y ~ and From (4.2), this contradicts the global optimality of y in problem (Py f ). Therefore y* must belong to the boundary of y s (b) From the discussion in Section 3.2 s ince Y ~ is a nonempty convex set and, for each j = 1 2, ... p, the function h i ( y) = y i is positive and concave on Y ~, the global optimal s olution se t for problem ( Pr f ) is identical to the global optimal solution set for the problem where for each y E Y ~ g : Y ~ R is the concave function defined by 1/p Since y s is a nonempty clo se d convex set with at least one extreme point from Rockafellar ( 1970 Corollary 18.5.3), it is easy to see that Y ~ can contain no lines ,.. Further1nore s ince problem ( P rf) has at least one global optimal solution problem ( P rs)
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82 also bas at least one global optimal solution. By Rockafellar (1970, Corollary 32.3 1), since g is a concave function on Y s; the latter two statements imply that problem (PY~) has at least one global optimal solution y that is an extreme point of Y ~ Because the optimal solution sets of problems (Pr ~ ) and (PY ~ ) coincide, this completes the proof. Suppose that Y ~ is a nonempty, closed convex subset of R !, and that y f contains at least one extreme point. Then, from Theorem 4 3.2, there will exist at least one global optimal solution for problem (PY ~ ) that is an extreme point of Y ~ and all global optimal solutions for problem (Pr ~ ) will lie on the boundary of Y ~ Neither of these properties however is necessarily shared by the decision setbased problem (Px) whose outcomespace reformulation yields problem (PY ~ ). The following example demonstrates this and in problem ( P x ). Then X is a nonempty, convex set and for each i = l, 2, / ; is a convex, positively va lu e d function on X Therefore, by Theorem 4.3.1, the problem (Pr ~ ) obtained by formulating the outcomespace version of problem (Px) is guaranteed to satisfy the conditions of the convex case for problem (PY ~ )Furthermore, it is not difficult to show in this case, that Y ~ is compact. Thus, Y s; is closed and contains at
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83 least one extreme point. It is easy to see that the unique global optimal solution to problem (P r~ ) is (y f = (1, 1) which, as guaranteed by Theorem 4.3.2, is an extreme point of y s (and is thus on the boundary of Y ~ ). On the other hand, the only global optimal solution to problem (P x ) is (x f = (1, 2 ), yet x is neither on the boundary of X nor is it an extreme point of X. To present the next result, we need to define two types of functions. Definition 4.3.1. Let Z R 11 be a nonempty convex set, and let h: Z R. The function h is said to be guasiconcave on Z when for each z', z 2 e Z and A e R such that o
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84 Theorem 4.3.3. Suppose that problem (P Y~ ) satisfies the conditions for the convex case. Then, in this problem, g is a strictly pseudoconcave function over the convex set y < Proof. The set y < is a convex set by definition of the convex case for problem (PY ~ ). To show that g is strictly pseudoconcave over Y ~ notice first that by ( 4.2), g can be considered to be well defined over the open set R:. Also notice that g is differentiable over R : and, thus over Y ~ k R : Suppose now that y 1 and y 2 are distinct points in Y ~ that satisfy (v g(y 1 ), y 2 y 1 ) 0 Then from ( 4.2), we obtain o ~( v g ( y1 ), y2 Y 1) = f IJ y~ y ; y!) k = I j'#k =t k= l j= I By multiplying both sides of ( 4 17 ) by (1 / p) and rearranging, we obtain that g ( y ) ~ g(yl )( 1 / P )t (y ; / Y! ). k= l From Lemma 4.2.1 ( 4.17) ( 4.18 ) ( 4.19)
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85 with equality holding iff, for some M > 0, y; =MY!, k = 1, 2, ... p. There are two cases to consider. Case (i): There is no M > 0 such that Yi =MY!, k = 1, 2, ... p. Then, in ( 4.19), strict inequality holds, so that from ( 4.18) and ( 4.19), Since g(y')>O, this implies that g(y 2 )< g(y 1 ). Case (ii): For some M > 0, Yi= MY!, k = 1, 2, . p. If we choose such an M, then (4.19) holds as an equality. Thus, from (4.19) and the choice of M, we obtain that 1/p g(yl )(1/ p )L (y; I Y! )= g(yl )~(y 2 )/ g(yl )] (4.20) k= l and that (4.21) respectively. Since g(y 1 )> 0, together (4.18), (4.20) and (4.21) imply that Dividing through by g(y 1 )> 0 yields M ~1. Notice that M "I= 1, since, by assumption, y 1 and y 2 are distinct. Therefore M < 1. By (4.21), since g(y 1 ), g(y 2 )> 0, this implies that g(y 2 )< g(y 1 ), and the proof is complete. Remark 4.3 1. Theorem 4 3.3 justifies and strengthens the claim of Sniedovich and Findlay (1995, p. 317) that when Y ~ is a convex subset of R!, g : Y ~ R defined by ( 4.2) is differentiable and pseudoconcave on Y ~
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86 From Theorem 4.3.3, in the convex case, problem (Pr ~ ) is a global optimization problem involving the minimization of a strictly pseudoconcave function over a convex set Y ~ Therefore, as in the general case, multiple local optimal solutions for problem (P s ) will generally exist that are not globally optimal. Y From Theorem 3 2.1, we know that when Y ~ is a nonempty, convex subset of R:, the function g: Y ~ R defined, as in the proof of Theorem 4.3.2, by g(y) = [g(y )]1 /p (4.22) is concave where g : f ~ R is given by (2). By the next result, when the domain of g is restricted to an appropriate subset of Y ~, a stronger statement can be made. Theorem 4.3.4. Assume that Y ~ is a nonempty, compact, convex subset of R:. For any ae RP and beR such that a>O and b>O, let Z(a,b)=Y ~ n {yeR P( a,y)=b}. Then g: Z(a,b) R defined for each ye Z(a,b) by (4.22) is a strictly concave function for any a e RP and be R such that a > 0 and b > 0. Proof. Assume that y 1 y 2 e Z(a,b) and y 1 :t y 2 where ae RP, be R, a> 0, and b > 0. Since Z(a,b) is an intersection of two convex sets, it is itself a convex set. Therefore, if we choo s e A e R such that O < A < 1 then z :=Ay 1 +(1A)y 2 e Z(a,b). Also, by (2) and ( 4.22 ), 1 /p g(z)= IT [;t, y~ + (1A )y ~ ] (4.23) j= I From Polya and Szego (1972),
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87 p 1 /p p fI [1 y~ + (1A)y ~ ] AFIJ y~ I / p p + (11)"IJy } 1 / p (4.24) j = l j= I j=l with equality holding iff A y~ = K(1A )y ~, j = 1, 2, .. p, for some positive constant K. Since and 1/p = (1A )g(y 2 ), (4.23) and (4.24) will imply the desired result if we can show that no K > 0 exists such that A y~ = K(l A )y ~, j = l, 2, ... p. (4.25) Notice that since y 1 =Iy 2 K ;. := [1 / (1A)] does not satisfy ( 4.25). Suppose, to the contrary that for some K > 0, (4.25) is satisfied. Then from ( 4.25) it follows that y' = K[(11)/1]y 2 (4.26) Since y 1 y 2 E Z(a,b ), (4.27) Substituting for y 1 in ( 4.27) via ( 4 26), we obtain K[(l A)/ A]( a, y 2 ) = ( a, y 2 ) = b.
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88 Solving here for K, we obtain that K = [i/(1A)]. Since K = K.,. = [i/(1A)] does not satisfy ( 4.25), this contradiction concludes the proof. It is important to notice that the counterpart of Theorem 4.3.4 in the decision space does not hold, even in the polyhedral case. In particular, suppose that X k R n is a nonempty, compact polyhedron and, for each j = 1, 2, ... p, that there exists a c i e Rn such that J i (x) = ( c i, x) > 0 for all x e X. Then, although the function h : X R defined for each x e X by 1 /p p h(x)= I1 (ci' x) j=1 is concave (see Theorem 3.2.1), the function h: X(a,b ) R need not be strictly concave, where a e R P, be R a> 0, b > 0, and X (a b) = x e X f a i ( c 1 x) = b j=l The foil owing example illustrates this observation. Example 4.3.2. Let (4.28) and let J i (x 1 x 2 ) = ( (1, 1 ), (x,, x 2 )), j = 1, 2. Then X is a nonempty, compact polyhedron and, for each j = 1 2, Ji is po s itive and linear on X As guaranteed by Theorem 3.2 1, h: X R, which by ( 4.28 ), i s given by
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89 is concave However, if for example, a 1 = a 2 = 1 and b = 4, then h is not strictly concave on Con s ider now problem ( Py ~ ) when the conditions of the polyhedral case hold Assume also that Y ~ i s a compact set and that y E Y ~ For algorithmic purposes, it may be quite useful in thi s case to develop tools for finding local optimal s olutions for problem ( P r ~ ). The s e tools could then potentially be used to construct global s olution algorithms for the problem that repeatedly move from a local optimal solution to an improved local optimal solution until a global optimal solution is found. The remaining results in thi s section are motivated in part by the de s ire to find such tool s. Notice th a t in the polyhedral case the optimization problem in (9 ) i s a linear program given by p ( LP ) min (1 / p )L ( y j /y j ) s t. y E Y ~ j= I Problem ( LP ) will have an optimal solution y* that can be found, for instance by the simplex method Since y E y < the minimum value vmin in problem (LP ) sati s fies vmin 1.0. As a result, ther e are three pos s ible ca s es for problem (LP). First, vmin < 1.0 may hold. Second vmin = 1.0 may hold with y being the unique optimal solution to problem ( LP). Third vmin = 1 0 may hold, with problem ( LP ) having multiple optimal solutions In the fir s t case from Theorem 4 2 2 it follow s that g ( y ) < g( y ) wher e y* i s any optimal solution t o problem ( LP ), so that a more attractive feasible s olution y to
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90 problem (Pr ~ ) than y has been found. To analyze the second case, we need the following two definitions and lemma. Definition 4 3 3. A point y E Y ~ is a strict local optimal solution for problem (PY ~ ) when there exists an E > 0 such that for each y E y s for which y y and y y < e, g(y )> g(y). Definition 4.3.4. Let Z be a nonempty convex set in Rn, and let h: Z R The function h is said to be strongly quasiconcave on Z when for each z 1 z 2 E Z with z' z 2 we have for each A such that O
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91 and that y = y i s th e unique optimal s olution to problem ( LP ) Then y i s a strict local optimal solution f o r problem ( P r i ). Proof Sinc e g ( y ) > 0 and y = y i s the unique optimal solution to the problem y = y must al s o be the unique optimal solution to the problem p mi~ g ( y ) (1 / p) L ( y j I y j ) } eY J = I Therefore, by Lemm a 4 2.1, y = y i s the unique optimal solution to the problem min(l / p ) ( V g( y ), y) yeY 1 Since (1 / p) > 0 a nd ( V g ( y ) y) i s a con s tant thi s implies that y = y i s the unique optimal solutjon to th e problem min ( V g ( y ) y y). y eY ~ ( 4.29 ) Therefore the optimal value of problem ( 4.29 ) i s 0 and for all ye Y ~ s u c h that y :;:. y, ( V g ( y ), yy) > 0 ( 4 30 ) Let d 1 d 2 . d k repre s ent th e directions of the edges of y s emanating from the extreme point y o ff & From ( 4 30 ), for all i = 1 2 ... k. By Theor e m 4.1.2 in B az araa Sherali, and Shetty ( 1993 ) this implies that there exi s t po s iti ve real s 8 ; i = l 2 ... k s uch that ( 4.31)
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92 for each A ; E (o, o j ) Let 8 = 1 / 2 min { o i l i = l 2, ... k }, and consider the points y and &l l &l 2 &l k y+ y+ .. y+ Then by definition of 8 and (4. 31 ), g ( y +&Ji)> g(y) ( 4.32) for each i = 1 2, . k. Let z be any element of the convex hull of y, y + &1 ;, i = 1 2, ... k such that z :t y and, for each i = 1 2, ... k, z :t y +&J i. Since g is a strictly pseudoconcave function on Y ~ it is al s o a strongly quasiconcave function on Y ~ (Bazaraa, Sherali Shetty 1993 ). As a result by Lemma 4.3.1, g(z ) > min {g( y1 g(y +&Li), i = 1, 2, .. ,k }. ( 4.33 ) From (4.32) and ( 4 33 ), g(z)> g(y). Since 8 > 0, this implies that there exists an e > 0 sufficiently small so that if ZE Y s lz y <, and z :t y, then g(z)> g(y). Under the assumptions of Theorem 4.3.5 if vmin = 1.0 but y = y i s one of two or more optimal solutions to problem ( LP ), then y need not be a strict local optimal solution for problem ( P r ~ ). The following example illustrates this point. Example 4.3 3. Let and let yr= (4, 4 f. Then y < i s a nonempty compact polyhedron in R ; and the assumptions of Theor e m 4 .3. 5 are s atisfied In this case, y E y s and y i s an optimal solution to problem ( LP). However, since (y 8 ) = (4+ 8 4of E y s and g(y 8 )< g(y) for all values of 8 such that O < 8 3, by Definition 4.3.3, y is not a strict local optimal
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93 solution for problem (Pr s ). (In fact, y is not even a local optimal solution for problem (Py f ). ) Notice that problem (LP) in this case has multiple optimal solutions. In the third case of problem (LP), vmin = 1.0 and problem (LP) has multiple optimal solutions. In this case, by the next result, as in the first case, an improved feasible solution for problem (Pr s ) is at hand. The proof of this result relies crucially on Theorem 4.3.4. Theorem 4.3.6. Assume that problem (Pr f ) satisfies the conditions for the polyhedral case, and that Y ~ is compact. Suppose that y is an optimal solution for problem (LP), and suppose that problem (LP) has multiple optimal solutions. Then, for any y :;= y that is an optimal solution for problem (LP), g(y )< g(y) must hold. Proof. Let y :;= y be an optimal solution to problem (LP) Then, since g(y)> 0, y is also an optimal solution for the problem p min g(y) (1/ p )L (y j /yj ), s.t. y E Y ~ j=I By Lemma 4.2.1, since y E y s this implies that y is an optimal solution to the problem min (1/ p )(V g(y ), y ), s.t. y E Y ~ Since (1/ p )(V g (y ), y) is a fixed number, it follows that y is an optimal solution to the problem min (1/ p )(V g(y), yy), s.t. y E Y ~ (4.34) By assumption, y is an optimal solution for problem (LP). Therefore, the optimal value of problem (LP) equals 1.0. From Theorem 4.2.3, this implies that the optimal value of
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94 problem (4.34) equals 0. As a result, since y* is an optimal solution for problem (4.34), (v g(y), y y) = 0. By the choice of y*, it follows that y*(LP)k{ye Yj(Vg(y),yy)=o}, (4.35) where y (LP) denotes the optimal solution set for problem (LP) From Theorem 4.3.4, since V g(y) > 0 and (V g(y), y) > 0, the function g defined by (4.22) over the set y := (y ~ n {ye RP (V g(y), y) = (V g(y), y) }) is strictly concave. Notice, in addition, that g is differentiable on Rt. By ( 4.35), y and y* both belong to r. From Bazaraa, Sherali, and Shetty (1993), since y* :t y, the latter three sentences together imply that From (2) and ( 4.22), this implies that ~(y )]1 /p < [g(y)]l /p + (1/ p )i:k(y)l / p /yj](y; yj) j = I = [g(y)]l / p + k(y)l / p Ip ]t[(y;yj)/yj] j = I = [g(y)]l /p + [g(y)][ ( 1 p)/p](1/ P )(v g(y), y Y) ( _)l / p = g y where the last equation follows from the fact that y e y. As a result, g (y ) < g (y). Remark 4.3.2. Suppose that problem (Pr ~ ) satisfies the conditions for the polyhedral case and that Y 5: is compact. Then, by using Theorem 4.3.6 and the discussion preceding it, a
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95 finite guaranteed method can be described for finding a strict local optimal extreme point solution for problem ( PY ~ ) For instance, one such method is as follows Algorithm 4.3 1. Strict Local Optimal Extreme Point Solution Search Initialization Step. Find an initial extreme point y of Y ~ Step 1. Find the optimal value vmin for linear program (LP). If vmin < 1.0, go to Step 2. If vmin = 1.0 go to Step 3. Step 2. Find any optimal extreme point solution y for problem (LP). Set y = y and return to Step 1. Step 3. If y i s the unique optimal solution for problem (LP), then stop: The point y is a strict local optimal solution for problem ( Py ~ ) If problem (LP) has multiple optimal solution s, then continue. Step 4. Find any optimal extreme point solution y "# y for problem (LP). Set y = y and return to Step 1 The method is finite because Y ~ must have a finite number of extreme points. The methods of Ben s on ( 1998d ) and Benson and Sun ( 1998) can be shown to be suitable for executing the initialization step and for solving the linear program (LP) in Steps 1, 2 and 4. Notice that the counterpart to this method for the decision setbased problem (Px) is not guaranteed to succeed This is shown by the following example. Example 4.3.4. Let X = {(x 1 ,x 2 )r E R 2 0 5~x 1 x 2 ~3.0, x 1 + x 2 ~2.0], and, for each x E R 2 let
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96 J 1 ( x )=x 1 +x 2 J 2 (x) = 4x 1 +4x 2 Then X is a nonempty, compact polyhedron, and, for each j = l 2 J i ( x ) is a linear function with po s itiv e values for all x EX If we generate the initial extreme point (x 1 f = (1 5, 0 5) of X for example then the linear program 2 min (1 / 2 )L [r i ( x ) / J i (x )] s.t. x E X j=I has optimal value 1 0 and it ha s exactly two extreme point optimal solution s These are x 1 and (x 2 r = (0 5 l 5) Sin c e f j ( x 1 )= f j ( x 2 ), j = l 2 the counterpart procedure to the above method w o uld in thi s case cycle without tennination between g enerating the extreme point x 1 and generating the extreme point x 2 Neither of these extreme points is a strict local optimal s olution for problem (P x ), where the definition of a s trict local optimal s olution x f o r probl e m ( P x) i s g iven by Definition 4.3.3 with x, X ( P x ), x and IT1 i repla c in g y, Y ~, ( P r~ ), y and g, re s pectively j = I 4.4. Discussion The analy s i s o f the out c omes pace problem fortnulation ( P r i ) of the multiplicative programming problem ( P x) yield s s everal results. The key o ne s are a s follows. First sin ce globally s olving problem ( P r i ) al s o essentially globally s olves problem ( P x), and s in c e Y ~ ge nerally lie s in a significantly smaller space than X there could be great comput a tional ga in to b e d e rived by con s tructing global opt i mization
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97 algorithms for problem (Px) that work directly 011 problem (Py ~ ) instead of on problem (Px). Second, the potential to create global solution algorithms for problem (Pr ~ ) is quite high. For instance, in the convex case where Y ~ is closed and has at least one extreme point, the analysis has shown that problem (PY ~ ) possesses at least one global optimal solution that is an extreme point of Y ~ This result could potentially allow researchers to create algorithms for solving problem (PY ~ ) that concentrate on searching among the extreme points of y < in ways similar to those used in existing global optimization algorithms for other non convex programming problems (Horst and Tuy 1993). Third, it appears potentially possible to construct global solution algorithms for problem (PY ~ ) that are based, at least in part, upon local optimal solution searches. Indeed, the analysis shows, for example, the potential to create search mechanisms for finding strict local optimal solutions for problem (Pr ~ ) in the polyhedral case if Y ~ is bounded Combined with ideas from global optimization such as relief indicator functions or cutting planes (Horst and Tuy 1993), this suggests that global solution algorithms for problem (PY ~ ) might be possible wherein successive local optimal solutions of smaller and smaller objective function values are found until a global optimal solution is found and the search tenninates.
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CHAPTERS AN OUTCOMESPACE CUTTINGPLANE ALGORITHM FOR LINEAR MULTIPLICATIVE PROGRAMMING 5.1. Introduction The linear multiplicative programming problem may be written p ( P x) minf(x) = Il ( c j, x ), s.t. xeX, j=l where X k R n i s a nonempty compact polyhedron p 2 is an integer, and for each j=l,2 .. p c jE R n s ati s fie s (cj ,x )> O for all X E X. For each j = l 2 ... p let y j E R sati s fy and let y E R P denote the vector with jth entry equal to y j j = 1 2, ... p Let C denote the p x n matrix who s e jth row equal s c;, j = 1 2 ... p. Recall from Chapter 1 that one of the more direct reformulation s of problem ( P x) as an outcomespace problem is given by ( P r ~ ) min g ( y )= Ii yj, s .t y E y s, j=l where Y ~ = { y E R P C x y y for some x E X }. Closely related to this refor1r1ulation i s the problem 98
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99 p (Py) min g(y )= IJ yi, s.t. ye Y, j= I where Y={ye R P y=Cx forsomexe xJ. In this chapter we develop an outcome s pace, cuttingplane algorithm for globally solving the linear multiplicative programming problem (Px ). To accomplish this, we use the framework of a pure cutting plane, decision setbased concave minimization method of Horst and Tuy ( 1993, pp 175184). We show how to adapt this method to solving the outcomespace formulation ( P Y ~ ) of problem (Px) for a global, extreme point optimal solution. Because p i s almo s t always smaller than n, often by several orders of magnitude, we expect that potentially considerable computational savings could be obtained by using the new outcomespace, pure cutting plane algorithm in s tead of a decision setbased approach. As a further computational enhancement, we also show that for purposes of implementation, the mechanics of the outcomespace, cuttingplane algorithm can be applied to the s maller problem ( Pr) instead of to problem (PY ~ ). The next sec tion gives so me theoretical prerequisites that will help to present and justify the steps of the new algorithm. The new outcomespace, pure cutting plane algorithm for globally so lving problem ( P x) is presented and analyzed in Section 5 3. Section 5 4 shows that to further enhance computational efficiency, in practice the new algorithm can b e applied to the outcomespace reformulation (Py) instead of to problem ( PY ~ ). A sample problem is globally so lved with the new algorithm in Section 5 5, and some concluding remarks are g iven in the la s t section.
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100 5.2. Theoretical Prerequisites The outcomespace algorithm uses extreme point search and cutting planes to find a global optimal extreme point sol ution to problem (Pr ~ ) From this solution, a global optimal extreme point solution to the original multiplicative programming problem (Px) can be easily recovered, as we shall see. The approach of the new algorithm relies on several theoretical results. In this section, we review or develop the necessary results of this type. The theoretical prerequisites concern problems (Px) and (Pr ~ ) and their relationships to one another. Before presenting these prerequisites, we must give some preliminary definitions. Definition 5.2.1 Let W be an open set in R n that contains Z ~Rn, and let h: W R. The function h is called strictly pseudoconcave over Z when h is differentiable over Z and, for each pair of distinct points z 1 z 2 E Z, if (Vh(z 1 ), z 2 z 1 ) 0, then h(z 2 )< h(z 1 ). From Bazaraa, Sherali, and Shetty (1993), a strictly pseudoconcave function h: Z R defined over a convex set Z is both pseudoconcave and quasiconcave over Z. The converse to this statement, however, is not true. For details, see Bazaraa, Sherali, and Shetty (1993). Definition 5.2.2. (Bazaraa, Sherali, and Shetty 1993). A point y E Y~ is called a strict local optimal solution for problem (Pr ~ ) when there exists an > 0 such that for each y E Y$ for which y :t= y and yyj < g(y) > g(y). Each of the next five results either is taken directly from or follows easily from results in Chapters 3 and 4. For each result, the appropriate reference is given.
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101 Proposition 5.2.1 (Theorem 4.3.1). The feasible region Y ~ for problem (Py :; ) is a nonempty polyhedral subset of R t := {y E R P y > 0 J Proposition 5.2.2. (Proposition 3.2 2). Problem (Px) possesses a global optimal solution that is an extreme point of X. Theorem 5.2.1. (T heorem 4.3.2). Problem (Py s ) possesses a global optimal solution that is an extreme point of Y ~ Theorem 5.2.2. (Theorem 4 .2. 1 ). If y is a global optimal solution for problem (Py :; ), then any x E X such that ex ~ y is a global optimal solution for problem (Px ). Furthermore the global optimal values of problems (Px) and (Py:;) are equal. Theorem 5.2.3. (T heorem 4.3.3 ). The objective function g of problem (Py ) is a strictly pseudoconcave function over R t Notice also in Proposition 5 .2 .1 that Y ~ is full dimensional and bounded. Additionally notice in Theorem 5.2.2 that given y* as defined there, for any x E X such that ex~ y, x not only is a global optimal solution for problem (Px ), but it also satisfies ex= y *. Together with Theorem 5.2.2 and an observation given in Benson and Sun (1998), this implies that if y* is a global optimal solution for problem (Pr s ), then the linear program min fu ; i= l s. t. ex u = y *, u 0, XE X,
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102 will have at least one optimal solution, and for any optimal basic solution (x, u )= (x ,0 )e Rn +p to this linear program, x is an extreme point global optimal solution for problem (Px ). Thus, given a global optimal solution for problem (Py l! ), a global optimal extreme point solution for problem (Px) can be easily recovered by solving a single linear program for a basic optimal solution. Based upon Theorem 5 2.1, the outcomespace, cuttingplane algorithm to be presented confines its search to extreme points of the nonempty, compact polyhedron Y ~ In fact, the algorithm searches only within a certain subset of the set of extreme points of Y ~. This is justified in part by the following result Theorem 5.2.4. Any point that is a global optimal solution for problem (Py ~ ) must also be an extreme point of y f and a strict local optimal solution for problem (PY ~ ). Proof. Either (i) problem (PY ~ ) has a unique global optimal solution or (ii) problem (P r ~ ) has multiple global optimal solutions. Case (i): Problem (Pr ~ ) has a unique global optimal solution. Let y represent this solution Then, by Theorem 5.2.1, y is an extreme point of Y :;;; Since y is the unique global optimal solution for problem (Py s ), g(y )> g(y) for all ye y s such that y i= y. Therefore, for all > 0 and for each y E Y :;;; such that y i= y and I yYI < g(y )> g(y). By Definition 5.2.2, this completes the proof for this case Case (ii): Problem (P y ~ ) has multiple global optimal solutions Let y be any nonextreme point of Y ~. Consider the linear programming problem
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103 min (1/ p )f (y j /y j } s t. y E Y ~. (5.1) j = I Notice that since y E y s, the optimal value v of ( 5 .1) satisfies v 1. Therefore, either v < 1 or v = 1. If v < 1 then, from Theorem 4.2.2, there exists some y E Y ~ such that g (y) < g (y). Therefore, in thi s case, y is not a global optimal solution for problem If v = 1 then since y is a nonextreme point of Y ~ the linear program (5.1) has multiple optimal solutions, including y. In this case by Theorem 4.3.6, for any y* :;; y that an optimal solution for (5 1), g(y* )< g(y) must hold. Therefore, in this case, y is not a global optimal solution for problem (P Y~ ). Since either v < 1 or v = 1, the arguments above imply that no nonextreme point of y s can be a global optimal s olution for problem (PY ~ ). Therefore, each of the global optimal solutions for problem ( Py s ) must be an extreme point of Y ~ Let y be an extreme point optimal solution for problem (Pr s ). Then, since the number of extreme points of y s is finite (see Murty (1983), for e.g.), there must exist an > 0 such that if y y < y :;; y, and y E Y s then y is not an extreme point of y s Choose such an and let y '# y satisfy y y < y E y s Then y is not an extreme point of y s Since each global optimal solution for problem (Pr i. ) must be an extreme
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104 point of y s, this implies that g(y) > g(y). As a result, by Definition 5.2.2 y is a strict local optimal solution for problem (PY ~ ), and the proof is complete. Recall that a point z in a set Z k R i s called an isolated point of Z when it is not a limit point of Z. From Theorem 5 .2. 4, we obtain the following result as an immediate consequence Corollary 5.2.1 Let (y s ) denote the set of global optimal solutions for problem (Pr ~ ). Then every point in (y s ) is an isolated point of (y ~ ). To conclude this section we state the following result. The proof of this result is immediate by definition. Proposition 5 .2 .3 Let y be an extreme point global optimal solution for problem (PY ~ ). Then, for each extreme point y E y s that i s adjacent to y, g(y) g(y) must hold 5.3. OutcomeSpace, CuttingPlane Algorithm To adapt the approach of the decision se tbased HorstTuy concave minimization algorithm ( Hor st and Tuy (1993), pp. 175184 ) to solving the outcomespace for1nulation ( P Y ~ ) of problem ( P x ), three tasks need to be iteratively executed. The first i s a certain strict local optimal solution search, the second involve s the construction of a cutting plane and the third i s a termination test. Before giving a formal statement of the new outcomespace algorithm, we will describe how the algorithm executes each of these three tasks
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105 5.3.1. Strict Local Optimal Solution Search Motivated by Proposition 5.2.3 and Theorems 5 2.1 and 5.2.4, the strict local optimal solution search seeks an extreme point y of Y ~ that is a strict local optimal solution for problem (Pr ~ ) and satisfies g(y )~ g(y) for each extreme point y of Y ~ that is adjacent to y. This search relies heavily upon repeatedly solving the linear program min (1/ p )f (y i /Y i ), s.t. ye y < (5.2) j= I as y is set equal to the values of various extreme points of Y ~ The linear program (5.2) was first proposed and studied in Chapter 4. The next three results for problem (5.2) follow immediately from Chapter 4. For each result, the appropriate reference is given. Theorem 5.3.1.1. ( Theorem4.2.2). Assume that ye Y s If the optimal value of linear program (5.2) is less than 1.0, then g(y )< g(y) for any optimal solution y of problem (5.2). Theorem 5.3.1.2. (Theorem 4.3.5). Assume that _y E y s If y is the unique optimal so]ution to linear program (5.2), then y is a strict local optimal solution for problem Theorem 5.3.1.3. (Theorem 4.3 6). Assume that y E f ~ If y is an optimal solution for linear program (5.2) and this linear program has multiple optimal solutions, then for any optimal solution y :/= y for problem (5.2), g(y )< g(y) must hold.
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106 From these results, we obtain the following procedure for finding a strict local optimal extreme point solution y E Y ~ for problem (PY ~ ) that satisfies g(y) g(y) for each extreme point neighbor y to y in y s Algorithm 5.3.1.1. Strict Local Optimal Solution Search Procedure. Step 1 Find an initial extreme point y of y s Step 2. Compute the optimal value v for linear program (5.2). If v < 1 0, go to Step 3. If v=l.O, go to Step 4 Step 3. Deter1nine any optimal extreme point solution y for problem (5.2). Set y = y* and return to Step 2. Step 4. If problem ( 5.2 ) has multiple optimal solutions, go to Step 5. Otherwise, determine whether or not g ( y ) < g (y) for some extreme point neighbor y to y in y s If so, find any such neighbor y, set y = y, and return to Step 2. If not, then stop: The point y is a strict local optimal extreme point solution for problem (Py ) that satisfies g(y )~ g(y) for each extreme point neighbor y to y in Y ~ Step 5. Find any optimal extreme point solution y for problem (5.2) that is distinct from y. Set y = y and return to Step 2. By using Proposition 5.2.1 and Theorems 5.3.1.15.3.1.3, it is easy to see that this search procedure is guaranteed to be finite and to find a strict local optimal extreme point solution y for problem ( Py ) s uch that g(y )~ g(y) for each extreme point neighbor y to y in Y ~
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107 5.3.2. Cutting Plane Construction Let y be a nondegenerate extreme point of Y ~ that is a strict local optimal solution for problem ( PY ~ ) and satisfies g(y) g(y) for each extreme point neighbor y to y in Y $ and let r E R satisfy r g (y). Given y the new algorithm will seek to find a vector re E R P such that (5.3) When re satisfies ( 5.3 ) the linear inequality (re, yy)~l is called a '.}':valid cutting plane (or '.}':valid cut) for problem (P Y ~ ) (Horst and Tuy 1993). Since y i s a nondegenerate, strict local optimal extreme point solution for problem ( PY ~ ) that satisfies g (y) g (y) for each neighboring extreme point y to y in Y s; since Y ~ has full dimension, and since g is strictly pseudoconcave on R! (cf. Theorem 5.2.3), it follows that an approach used by Horst and Tuy (1993, pp. 8591) can be used to find a vector re E R P that yields a yvalid cut. The following procedure applies this approach to problem (P Y ~ ). Algorithm 5.3.2.1. Procedure for Cutting Plane Construction. Step 1. Determine the p neighboring extreme points y;, i = 1, 2, ... p, to y in y s: Let I I l 2 u=yy,z=, ... p. Step 2. For each i E { I 2, .. p} such that (5.4)
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108 set z; = y +a i u ; where a ; eqt1als the value of the supremum in (5.4). For each i e {1, 2, ... p} such that (5.4) does not hold, let z; = y +a;u;, where a ; > 0 is a finite number chosen so that g(z ; )~ y. Step 3. Compute nr = er Q 1 where e ERP is the column vector of p ones, and Q is the p x p matrix with column i equal to (z ; y ), i = 1, 2, ... p, and stop: The linear inequality (n,yy)~l is a ,'Valid cutting plane for problem (Pr ~ ). Notice that in Step 1 of this procedure, since y is nondegenerate, the p neighboring extreme points described there are guaranteed to exist. Further111ore, because Y ~ has full dimension, u ; i = l, 2, .. p, are linearly independent (Murty ( 1983, pp. 138). Notice also in Step 2 that since y is a strict local optimal solution for problem (Py f ) for which g(y ; )~g(y) i=l,2, ... ,p, foreach i=l,2, ... ,p, either O
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109 Step 2' For each i E { 1, 2, ... s} such that (5.4) holds, set a 1 equal the value of the supremum in (5.4). For each iE {1, 2, ... ,s} s uch that (5.4) does not hold, let a 1 > 0 be a finite number such that g(y+a ; u;)~r. Step 3' Let 1C be any basic solution to the system ( 5.5) and stop: The linear inequality ( 1e, yy)~ l is a yvalid cutting plane for problem (PY ~ ). Since y s ha s full dimen s ion and y is an extreme point of Y ~, by logic similar to that used in the proof of Lemma ID 1 in Horst and Tuy ( 1993 ), it can be shown that in Step 3', (5.5) has at least one basic solution. Further1nore, by showing a result similar to Proposition ID 1 in Horst and Tuy ( 1993) the modifications to the cutting plane construction procedure described above can be shown to be appropriate for generating a yvalid cutting plane for problem ( P Y ~ ) when y is degenerate 5.3.3. Termination Test The linear programmingbased termination test for the outcomespace, cuttingplane algorithm is contained in the following result. Proposition 5 .3.3.1. Let y be an extreme point of ys that is a strict local optimal solution for probl em ( P r ~ ) and sa ti sfies g (y) g (y) for each extreme point neighbor y to y in ys_ Let ( 1e, yy)~ t
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110 be a yvalid cutting plane for problem ( PY ~), where g(y) r and, for some extreme point y 0 of Y ~, r = g(y 0 ). Then ( n yy) > I for all ye Y ~ such that g(y )< y. Hence, if 1 max ( n yy), s.t. ye Y ~ then y 0 is a global optimal solution for problem (Py ~ ). (5.6) Proof. Since (n, y y) 1 is a yvalid cutting plane for problem (PY ~ ), we know from (5.3) that ( n ,yy)~ l holds for all ye Y ~ such that g(y) 1 can be shown to hold for all ye Y ~ such that g(y )< y. It is easy to see that this implies that if (5.6) holds, then y 0 is a global optimal solution for problem (PY ~ ). 5.3.4. OutcomeSpace, CuttingPlane Algorithm By incorporating the strict local optimal solution search procedure of Section 5.3.1, the cutting plane construction procedure of Section 5.3.2, and the termination test of Section 5.3.3 into the framework of a decision setbased, cutting plane method for concave minimization proposed by Horst and Tuy (1993, pp. 175184), we obtain the following pure cutting plane algorithm for globally solving problem (Py f ). As shown in Section 5.2, from the global solution for problem (Py ~ ) obtained via this algorithm, a global optimal extreme point solution for problem (Px) can be easily recovered by solving a single linear program Algorithm 5.3.4.1. OutcomeSpace, CuttingPlane Algorithm.
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111 Initialization Step. By using the procedure in Section 5.3. l, find a strict local optimal extreme point solution y 0 for problem (Pr ~ ) that satisfies g(y) g(y 0 ) for each extreme point neighbor y to y 0 in y s Set r = g(y 0 ) and Y 0 = Y ~ Go to Step 0. Step k, k 0 Step k. l. By using the procedure in Section 5.3.2, with Y ~ = Yk, construct a y.valid cut Step k.2. Set n: = n: k, y = yk, and y s = y k in the linear program in (5.6), and compute a basic optimal solution yk to this linear program If (n:k, yk yk )~l, stop : The point y 0 is a global optimal extreme point solution for problem (Pr ~ ), and r is the global optimal value for problem ( P r ~ ) Otherwise, continue. Step k.3. Let Starting from yk, use the procedure in Section 5 3.1 to find a strict local optimal extreme point solution yk +I for the problem ( ) yk+I nung y, s.t. ye that satisfies g ( y) g ( yk+i ) for each extreme point neighbor y to yk+t in y k+i If g(y k+l )< r, then se t y = g(yk +I ) y 0 = yk + I and Y 0 = yk +I, and go to Step 0. Otherwise, go to Step k + 1.
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112 The following result and proof are guided in part by Theorem V .2 in Horst and Tuy (1993). Theorem 5.3.4.1. The outcomespace, cuttingplane algorithm for problem (PY ~ ) is finite when the sequence { n k } is bounded. If the algorithm stops, the point y 0 found by the algorithm is a global optimal extreme point solution for problem (Pr ~ ). Proof. The algorithm consists of a number of groups (or cycles) of steps. The beginning of a typical cycle of steps occurs when for some k 0, in Step k.3 the incumbent solution vector y 0 is set equal to yk + I, the incumbent value r is set equal to g(yk+l ), and y 0 i s set equal to yk + i. During the cycle of steps, one or more '}'valid cuts is added to Y 0 but the incumbent solution y 0 and the incumbent value r remain unchanged. The cycle ter1ninates when a point yk+t is found such that g(y k +t )< y. In view of Proposition 5.3.3.1 this implies that yk + i strictly satisfies all of the '}'valid cutting planes created thus far by the algorithm. As a result, yk+I is an extreme point of y s and it is distinct from all extreme points of Y ~ previously encountered. By Proposition 5.2.1 y s has a finite number of extreme points Therefore, the number of cycles in the algorithm must be finite. During a typical cycle one or more '}'valid cuts of the form (7th' y ,Yh) 1, where h 0 is an integer, is added to the current polyhedron Y 0 k y s. For a typical k 1, from Step h .3 of the algorithm, we know that
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113 ( 5 7 ) h = 0, 1 2, .. k 1 We also note that 1 ( k k k) } 0 = 'JC ,y y < ( 5 8) From (5.7 ) ( 5 8 ) and Corollary ill 2 in Horst and Tuy (1993 ) if { n } is bounded, then the length of the cycle mu s t be finite Since the number of cycles in the algorithm is finite, this implie s that when { n } i s bounded the algorithm must be finite. Now s upp ose th a t the a l g orithm stop s. Then for some k 0, with 'JC = n ", y = yk and Y ~ = y k, inequality ( 5.6 ) becom es satisfied after which the algorithm immediately stops. Furthermore a t the time the algorithm stop s, it has found at the beginning of the current cycle of s t e p s an extreme point y 0 of y f and of Y k that is the incumbent solution and s ati s fi es r = g( y 0 ) ~ g ( yk ) In addition, yk i s a strict local optimal extreme point s olution for th e problem min g ( y ) s. t y E Y k that satisfies g ( y ) g ( yk ) for e ach extreme point neighbor y to y k in Y k. Taken together the s e obser v ations imply that we may apply Proposition 5.3.3 1 with n ='!C k, y = yk, y 0 = y, a nd Y ~ = Y k t o conclude that y 0 is a global minimizer of g o ver y k If y E y < and y e y k, then Step k.3 implies that for some f valid cut ( n j y yj)~ l where f y ( n i yy j ) < 1 mu s t hold By ( 5 3 ), s ince r ~ f thi s implie s that g( y ) ~ r r Sin ce r = g ( y 0 ) it follow s that g ( y ) ~ g ( y 0 ) for all ye Y k LJ { ye Y ~ j y e y k },
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114 i.e., y 0 is a global optimal extreme point solution for problem (Pr ~ ) 5.4. Implementation In practice, by moderately modifying the procedures and the termination test in Sections 5.3.15.3.3 and by substituting Y for y f and the phrase ''problem (Py)'' for ''problem (Pr ~ )'' throughout the statement of the new outcomespace, cuttingplane algorithm we obtain an algorithm that globally solves problem (Py) instead of problem (P Y~). Since Y k y < computational savings can generally be expected to result by solving problem (Py) instead of problem (Pr ~ ). The validity of this approach relies in part upon the following two results. Proposition 5 .4.1. Let y be a global optimal solution for problem (PY ~ ). Then y e Y, and any x e X such that Cx y satisfies Cx = y is a global optimal solution for problem (Px ). Proof. Since Cx > 0 for all xe X, it is easy to see by the definition of Y ~ that the global optimality of y in problem (Py ~ ) implies that if Cx y* for some x e X, then ex= y*. As a result, y "' e Y, and, by Theorem 5.2.2, x is a global optimal solution for problem (Px ). A strict local optimal solution for problem (Py) is defined as in Definition 5.2.2, except with Y replacing Y ~ in the definition.
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115 Proposition 5 4 2. Problem (P y) possesses a global optimal solution that i s an extreme point of Y. Any g lobal optimal solution for problem (Py ) must also be a s tric t local optimal s olution for problem ( Py ) Proof From Rockafellar ( 1970 ), Y is a polyhedron Since X is nonempty and compact Y i s al s o nonempty and compact. Ft1rther1nore since Cx > 0 for all x e X, Y { z e R P z > 0 }. Therefore, from Proposition 3 2.2, problem (P y ) has a global optimal solution that is an extreme point of Y A proof that any global optimal solution for problem ( P y) mu s t be a s trict local optimal solution for problem ( P y) i s given by the proof of Theorem 5 .2 4 except with Y repla c ing y s in the proof. Remark 5.4.1. From Propo s ition 5.4 1 we may confine the search for a global optimal solution for probl e m ( P Y ~ ) to Y Stated in a different way, any global optimal solution to problem ( Pr ) i s al s o a global optimal solution to problem (Py ~ ). By Proposition 5.4.2, problem ( P y) ha s a global optimal s olution that is an extreme point of Y, and any global optimal extreme p o int s olution to problem ( P y) must be a strict local optimal solution for problem ( Py ) Taken together since Y is a nonempty compact polyhedron these observations imply that to globally s olve problem ( P y s ) we may apply the outcomespace, cutting plane approach given in Section 5 3 to problem (Pr), rather than to problem ( P Y ~ ). U s ing thi s approach we will find a global optimal solution to problem (P r s ). By Propo s ition 5 4.1 a g lobal optimal extreme point solution to problem ( P x) can then be recovered by s olving a s ingle linear program as explained in Section 5.2.
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116 In addition to s ubstituting Y for Y ~ and the phrase ' problem ( P r)'' for ''problem (P r~) '' throughout th e statement of the new outcomespace, cuttingplane algorithm in Section 5.3, some changes are needed to the procedures given in Sections 5.3.1 and 5.3.2 and in the termination te s t in Section 5 3 3 in order to confine the search to Y y s Let us now explain these change s. The following two re s ult s help to explain the changes needed in the s trict local optimal solution s earch procedure given in Section 5.3.1 to confine the search to Y. The proof of the fir s t r es ult is immediate from the definitions. Proposition 5.4.3 An y strict local optimal solution for problem (Pr ~ ) i s also a strict local optimal solution for problem ( P r). Proposition 5 4.4. As s ume that y E Y. Then any optimal solution for linear program ( 5.2 ) belongs to Y Proof Suppo s e that y i s an optimal s olution to problem (5.2 ), but y* Y. Then, since y E Y ~ but y Y there exists some point x E X such that Cx y* but C x "# y Let y ** = C x *. Then y E y s Further1nore, since y > 0, y y and y ** "# y it follow s that j=l j1 Since y "',. E y <, thi s in e quality contradict s that y"' is an optimal solution to problem ( 5.2 ) Therefore y Y cannot hold and the proof i s complete.
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117 From Propo s ition s 5 4 3 and 5 4 4, if we replace Y ~ with Y in problem (5.2 ) and throughout the strict local optimal solution search procedure of Section 5 3 .1, and if we replace the phrase ' problem ( P y ~ )' by ''problem (Py)'' in the procedure then the modified procedure i s guaranteed to find a strict local optimal extreme point solution y for problem ( P y) that s atisfie s g ( y )~ g( y ) for each extreme point neighbor y to y in f. Thus, to apply the lo c al search procedure in Section 5.3.1 to problem (P y), we need only to replace Y ~ with Y in linear program ( 5 2 ) and throughout the statement of the procedure and to change the phra s e '' problem (P y ~ ) '' to ''problem (Py)'' in Step 4 of the procedure. The cutting plane construction procedure given in Section 5.3.2 relies on the fact that at least p ed g e s of y < will emanate from every extreme point y of Y ~ where p is the dimension of Y ~ The dimension d of Y, however may be less that or equal to p. As a prerequisite to applying the cutting plane construction procedure when problem ( P y) is being solved instead of problem (Pr s ), a knowledge of the value of the dimension d of Y i s required A convenient way to find d is to insert the following step between Steps 1 and 2 of the strict local optimal solution search procedure given in Section 5.3 1 Step l Find all neighboring extreme points y; i = l, 2, ... t, to y in Y. Compute and s ave the rank d of the t x p matrix M, where row i of M is equal to I 1 2 yy l =, . t
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118 From Murty (1983, pp. 138), the value of d computed in Step 1' equals the dimension of Y. When d = p, the cutting plane construction procedure given in Section 5.3.2 can be directly used when solving problem (Py) by the new outcomespace, cuttingplane algorithm by simply replacing Y ~ by Y and replacing the phrase ''problem (Pr ~ )'' with ''problem (Py)'' throughout the procedure. When d < p, a recent generalized cutting plane procedure given in Benson (1999) can be used instead of the procedure given in Section 5.3.2. The termination test that is needed to adapt the outcomespace, cuttingplane algorithm for problem (PY ~ ) to problem (Py) is obtained by replacing Y ~ in (5.6) by Y. The validity of this change can be supported by proving a result for problem (Pr) that is analogous to Proposition 5.3.3.1. To implement the changes called for in this section to the new outcomespace, cuttingplane algorithm given in Section 5.3.4, mechanics must be available for finding an initial extreme point of Y and, given an extreme point y of Y, for finding all neighboring extreme points in Y to y Given these mechanics, one can then carry out all of operations called for by the revised outcomespace, cuttingplane algorithm, including the solution of all of the required linear programming problems over Y and the construction and incorporation into Y of all required '}'valid cuts. A finite linear programmingbased algorithm guaranteed to find an initial extreme point of Y is given in Benson (1998d). In Benson and Sun (1998), the mechanics are given for finding all extreme points of Y that are adjacent to a given extreme point of Y.
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119 From the previous paragraph, the existence of these procedures guarantees that the outcomespace, cuttingplane algorithm for globally solving problem (Py) can, indeed, be implemented in practice. As a result, as we have seen, a global optimal extreme point solution for problem (Px) can be found. 5.5. Example To illustrate the application of the outcomespace, cuttingplane algorithm to problem (Px ), consider the case with p = 2 and X = {xe R 11 Ax= b, x O }, where 9 9 2 1 0 0 0 0 0 0 0 81 8 1 8 0 1 0 0 0 0 0 0 72 1 8 8 0 0 1 0 0 0 0 0 72 7 I 1 0 0 0 1 0 0 0 0 9 A= b= 1 7 1 0 0 0 0 1 0 0 0 9 I I 7 0 0 0 0 0 1 0 0 9 1 0 0 0 0 0 0 0 0 1 0 8 0 I 0 0 0 0 0 0 0 0 1 8 and where I 0 0 I 1/9 1/9 0 0 0 0 Cl= 0 C2 = 0 0 0 0 0 0 0 0 0 0 0
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120 It can be shown that, in this case, X is a nonempty, compact polyhedron, and Cx > 0 for all xE X, as required in problem (Px ). To solve problem (Px ), as explained previously, we will apply the modified outcomespace, cuttingplane algorithm to problem (Py). In this case, problem (Py) is given by Xi+ (1/9 )~ X 2 + (1/9 )x 3 for somexe X. A summary of the application of the modified outcomespace, cuttingplane algorithm to problem (Py) in this case is given below. The mechanics used on the set Y, including the generation of an initial extreme point of Y and the solution of linear programming problems with feasible regions given by Y, are taken from Benson (1998d) and Benson and Sun (1998). Initialization Step. We execute the search procedure given in Section 5.3.1, but adapted to problem (Py). The initial extreme point y of Y that we find thereby is given by y 7 = (1/9, 8 1/9). As called for by Step 1' in Section 5.4, we find that the extreme point neighbors to y in Y are (y 1 r = (1, 1) and (y 2 f = (9/10, 8 1/10). The matrix M is thus given by M= 8/9 71/90 7 1/9 1/90 We set d = rank M = 2, and we save d = 2 for future reference. Next, with Y replacing Y $ in (5.2), the optimal value of linear program (5.2) is found to be v = 1.0, and we find that y is the unique optimal solution to this problem. Therefore, we compare
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121 g (y) = y 1 y 2 = 73/81 to the values of g evaluated at the neighboring extreme points y 1 and y 2 to y in Y. We find that g(y) = 73/81 < g(y 1 )= 1.0 and g(y)= 73/81 < g(y 2 )= 7.29. The strict local optimal extreme point search is therefore terminated, and we set y 0 = y, where yr =(1/9,8 1/9). We set r=g(y 0 )=73/81, Y 0 =Y, andproceedtoStepO. Step 0. Step 0.1. We construct a rvalid cut for problem (Py) with Y = Y 0 at y 0 We accomplish this by executing the procedure given in Section 5.3.2, but adapted to problem (Py) with Y = Y 0 In particular, we first deter111ine whether y 0 has exactly d or more than d neighboring extreme points in Y. In this case, we know from the initialization step that y 0 has exactly d = 2 neighboring extreme points in Y. Therefore, y 0 is nondegenerate, and we will execute the cutting plane procedure given in Section 5.3.2 for the nondegenerate case, modified for problem (Py). Towards this end, we set (u 1 ) = (8/9, 7 1/9) and (u 2 ) = (71/90, 1/90 ). Next, we compute sup {ei > 0 g(y + 0 ; ui )~ r }, i = 1, 2, and we find that for i = 1, this supremum is finite and equals 1.015625 and, for i = 2, this supremum is finite and equals 729.85915. Therefore, we set a 1 = 1.015625 and . a 2 = 729.85915. Then, for each i = 1, 2, we set z' = y +a;u', which results in the values
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122 (z 1 r = (1.013888, 0.888888), ( z 2 j = (575.888888, 0.001565). Next, we compute ( rc 0 f = er Q 1 where 0.902777 Q = 7 .222222 575.777777 8.109546 This yields (rc 0 ) = (0.0002138, 0.1384883). Theyvalid cut, after gathering terms, is then given by 0 0002138y 1 +0.1384883y 2 ~0.1233175. Step 0 2. With re= ( n f y = y 0 and Y ~ = Y 0 the linear program in (5.6) is found to have y 0 as a basic optimal solution, where (y 0 J =(73/9,1/9), and an optimal value of 1.1064. Since 1.1064> 1.0000, we continue to Step 0.3 Step 0.3 We first set Y 1 = Y 0 n{yE R 2 0.0002138yl + 0.1384883y 2 ~0.1233175}. Next, starting from y 0 we execute the search procedure given in Section 5.3.1, but adapted to problem (Py) with Y = Y 1 in order to find a strict local optimal extreme point solution y 1 for the problem min g(y ), s .t. y E Y 1 that satisfies g(y) g(y 1 ) for each extreme point neighbor y to y 1 in Y 1 This results in (f 1 r =(8 1 /9, 1/9).
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123 Since g (y 1 )= 73/81 r, we proceed to Step 1. Step 1. Step 1.1. We construct a rvalid cut for problem (Py) with Y = Y 1 at y 1 To do so, we use the procedure in Section 5.3.2, but adapted to problem (Py) with Y = Y 1 We find that y 1 has exactly d = 2 neighboring extreme points in Y 1 so that y 1 is nondegenerate. The resulting rvalid cut is, after gathering terms, 0.1384833y 1 +0.0002138y 2 ~0.1233175. Step 1.2. With n = n 1 y = y 1 and Y 5 = Y 1 we find that the linear program in (5.6) has the point y 2 given by ( y 2 ) T = (1.9003173, 0.8874603) as a basic optimal solution, and that it has an optimal value of 0.8599391. Since 0.8599391 < 1.0, the algorithm stops, since it has found the global optimal solution y 0 for problem (Py), where (y 0 ) = ( 1/9 8 1/9) and r = 73/81 is the global optimal value for problem (Py). As explained in Remark 5.4.1 and Section 5.2, by solving a single linear program, an extreme point global optimal solution to problem (Px) can be recovered from y 0 in the sample problem By solving this linear program, we obtain the extreme point global optimal solution (x) = (0 0, 8.0, 1.0) for problem (Px ).
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124 5.6. Concluding Remarks Both theoretical and empirical evidence have shown that generally an outcome polyhedron can be expected to have a significantly lower dimension, a significantly simpler structure, and far fewer extreme points than the underlying decision set polyhedron. For the case of the linear multiplicative program (Px ), we have in this chapter constructed and validated an outcomes pace, cuttingplane algorithm that searches the outcome polyhedron, rather than the decision set polyhedron for a global optimal extreme point solution. We expect, therefore, that the algorithm proposed here for problem ( Px) will potentially have computational benefits over the decision setbased algorithms that have been proposed for the problem.
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CHAPTER6 SUMMARY AND FUTURE RESEARCH 6.1. Introduction In this dissertation, we have analyzed the multiplicative programming problem (Px). For the linear case of problem (Px), we used these analyses to develop a heuristic algorithm that find s a good solution to the problem and to present a global solution algorithm for the problem. In this chapter we will examine potential avenues for future research that might potentially improve the efficiency and accuracy of the heuristic algorithm We also examine possible avenues for potential global optimal solution algorithms for problem ( P x) based on analytical results we have presented. 6.2. Future Research on the Heuristic Algorithm In preparation for presenting our heuristic algorithm for the linear case of problem (Px) in Chapter 3, we analyzed the case when problem ( Px) is the concave multiplicative programming problem We showed that when problem (P x ) is a concave multiplicative programming problem with a compact feasible region, it has at least one global optimal solution that is an efficient extreme point of the associated multiple objective programming problem ( MOMP ) VMIN J(x ) s.t. xe X where f j : X R j = 1 2 ... p, are the functions used in defining problem ( Px ) and where f ( x ) denote s the vector 125
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126 [J 1 (x ), / 2 (x ), . J P (x )]r. Our heuristic algorithm for the linear multiplicative program with a compact polyhedral feasible region, uses this efficiency result to search for a good solution by limiting the search to the efficient s et of the multipleobjective linear program ( MOLP ) VMIN C x, s.t xe X, where C is the p x n matrix whose jth row equals (c i ) r j = 1, 2, ... p. The heuristic algorithm first id e ntifies an efficient face X 1 11 of problem (MOLP). It then searches the efficient face X 111 for a n good s olution to problem ( P x) by optimizing a linear approximation of the objective function of problem ( P x ) over X w Our experimental results show that the c omputational requirement s of the heuristic algorithm are not overly burdensome when compared to the effort required to solve a linear multiplicative programming problem The performance of the heuristic algorithm depends in part upon the number location s, and dimen s ion s of the efficient face s found in the Efficient Point Search Phase. The user can manipulate the location and number of efficient faces found by selecting values for the parameter s N u se d for the weighting of the functions in the objective function and S th e number of s ample objective function values from Y ~. A topic for further research would be to identify criteria and methods for choosing good values for these parameters s o that a variety of regions of the efficient set are searched for a good solution for problem ( P x ) In Step 1 of th e Efficient Point Search Pha s e, an efficient extreme point is used to identify an effici e nt face X 111 on which that extreme point lies. Currently there is no
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127 known procedure that can be used to find an efficient face of maximal dimension on which an extreme point lies. Since the perf orrr1ance of the heuristic algorithm can be improved by searching maximal efficient faces for good solutio ns, a topic for further research would be to find such a procedure. 6.3. Future Research on an Global Solution Algorithms In Chapter 3 we showed that a concave multiplicative programming problem can be solved as a quasiconcave minimization problem or reformulated as a concave minimization problem. We also showed that a concave multiplicative programming problem with a compact feasible region has a globally optimal solution that is an efficient extreme point of the associated multipleobjective programming problem (MOMP). A potential opportunity therefore exists to improve the efficiency of a concave minimization method for solving the concave multiplicative programming problem or the linear multiplicative programming problem by limiting the search to the efficient extreme points of the associated multipleobjective programming problem (MOM P ) or (MOLP) respectively. For example, for the linear case of problem (Px), enumerative concave minimization methods such as extreme point ranking or cutting plane methods can be modified to limit the search to those extreme points of problem (MOLP) that are efficient. For the concave case of problem (Px), outer approximation and branch and bound methods may be modified to limit the search to efficient extreme points of problem (MOLP). For more information on concave minimization methods, see, for example, Benson (1995) and Horst and Tuy (1993).
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128 In preparation for developing our outcomespace, cuttingplane algorithm for the linear multiplicative programming problem in Chapter 4, we analyzed the outcomespace reformulation problem (Pr ~ ) of the multiplicative programming problem (Px). Since the reformulation (PY ~ ) works in a lower dimensioned space than the space of the feasible decision set X, potential computational savings can be expected by solving problem (PY ~ ) for an optimal solution y* E Y ~ As we have shown, a global optimal solution x* EX for problem (Px) can then be recovered from y*. In Chapter 4 we showed that when problem (Px) is a convex multiplicative programming problem, a global optimal solution y of the reformulation (PY ~ ) will exist on the boundary of the convex outcome set Y ~ In addition, when the convex set Y ~ is closed and has at least one extreme point, our analysis showed that problem (Pr ~ ) has at least one global optimal solution that is an extreme point of y s We also showed that the objective function of problem (PY ~ ) is strictly pseudoconcave over y s Since pseudoconcave functions defined over convex sets are also quasiconcave functions, many of the most popular algorithms for minimizing concave functions over a convex sets are equally suitable for minimizing quasiconcave functions over convex sets (Horst and Tuy 1993 and Benson 1995). Our results thus provide possible avenues for proposing implementable solution methods for the convex multiplicative programming problem that solve instead the reformulation (Pr s: ) using existing concave minimization methods. For example, outer approximation can be used to search the boundary of Y~ for a global optimal y* solution for problem (Pr s: ). This method cannot be applied directly to the
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129 original convex multiplicative programming problem since, as shown in Example 4.3.1, the preimage x EX of y need not lie on the boundary of X. For more info1111ation on concave minimization method s see, for example, Benson ( 1995) and Horst and Tuy (1993). In Chapter 5 we presented the outcomespace, cuttingplane algorithm for solving the case when problem (P x ) is the linear multiplicative programming problem It solves the reformulation problem (PY ~ ) of problem (Px) for a global optimal extreme point solution y E Y $ for problem (PY ~ ), and then recovers a global optimal extreme point solution x EX of problem (P x ) The refor1nulation (PY ~ ) of the linear multiplicative programming problem (Px) offers another computational advantage in addition to that of working in a lower dimensioned space than the space of the feasible region X A large number of extreme points of X collapse into nonfacial structures or into the relative interiors of subfa c e s of the outcome set Y ~ but not vice versa (Benson 1995a). Since the outcomespace, cuttingplane algorithm searches only those extreme points of Y ~ that are the image of extreme points and faces of X under the mapping of the functions in the objective function of problem ( Px), many nonoptimal extreme points of X are implicitly bypassed in the algorithm's search We therefore expect that the outcome space, cuttingplane algorithm will potentially have computational benefits over the decision setbased algorithms that have been proposed for the linear multiplicative programming problem. To aid in the implementation of the outcomespace, cuttingplane algorithm, the algorithm can be implemented using only linear programming methods. A topic for
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130 further research is to code the algorithm and test its perforn1ance against other algorithms for solving linear multiplicative programming problems.
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BIOGRAPHICAL SKETCH George Boger received his Bachelor of Science in Mathematics in 1973 from the University of Central Florida. Following graduation he worked in the federal civil service for NASA and the U.S. Navy. In 1992 he received a Master of Science in Operations Research from the Florida Institute of Technology and entered the Ph.D. program in the Department of Decision and Info11nation Sciences at the University of Florida. After graduation, George plans to work in academia, teaching and conducting research 137
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Harold P. Benson, Chairman Professor of Decision and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standard s of scholarly presentation and is fully adequate, in s cope and quality, as a dissertation for the degree of Doctor of Philosophy. ision and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. s o J. Vakharia A sociate Professor of Decision and formation Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standard s of scholarly presentation and i s fully adequate, in scope and quality, as a dis s ertation for the degree of Doctor of Philo s ophy. Richard L. Francis Professor of Industrial and Systems Engineering
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This dissertation was s ubmitted to the Graduate Faculty of the Department of Deci s ion and Infor1nation Sci e nce s in the College of Bu s iness Admini s tration and to the Graduate School and was accepted as partial fulfillment of the requirement s for the degree of Doctor of Philosophy December 1999 Dean, Graduate School
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