Title: Optima
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Title: Optima
Series Title: Optima
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Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: December 1995
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Volume ID: VID00048
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T


I


DECEMBER

MA


MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER


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ICo rnell Uivrst


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S2s NDEEMER99


Rice University Professor John Dennis Is New MPS Chairman


Our new chairman, John Dennis, is the
Noah Harding Professor at the Department
of Computational and Applied Mathematics,
Rice University, Houston, Texas, USA. His
research interest is practical methods for opti-
mization, in particular, parallel methods fbr
nonlinear optimization described by coupled
nonlinear simulations, and he has extensive
experience with industrial applications. The
1983 book, Numerical Methods for Uncon-
strained Optimization and Nonlinear
Equations, written with R.B. Schnabel is
now available in the SIAM series "Classics
in Applied Mathematics. "He has been active
within SIAM, served two terms on its Council
and chaired the SIAMActivity Group for
Optimization. He founded and served as first
Editor-In-Chiefof the SIAM Journal for
Optimization, and he has been a co-editor
ofMathematical Programming.
In the following article Dennis expresses his
beliefthat the optimization community has
not done a good marketing job but predicts
that the future of scientific computing is... '.,
to see an increase in the application ofoptimi-
zation. He also discusses the current state of the
Society and important issues for the future.
-KAREN AARDAL


AS engineers improve their
ability to simulate airflow
around a car body or the flow
of fluids in porous media,
they want to select decision
variables that at least improve
if not optimize the situa-
tion they are simulating. I think that the applica-
tion of homemade 'voodoo optimization' tech-
niques based on physical analogies of questionable
relevance to problems like the TSP, where there
are. i .. combinatorial techniques, is due
largely to scientists and engineers who need opti-
mization but who do not realize that we have
some good techniques, so they do it themselves. I
remember a comment from a physicist at the
workshop Bob Bixby ran a few years ago, which
Bob .- ll... the 'Great Texas TSP Shootout.' Ev-
eryone was to bring their software and examples to
the meeting and compete. The combinatorial op-
timization techniques dominated the physical
analogy codes in all the tests. The physicist com-
plained that, since the optimizers never published
in Scientific American, no one knew about all the
work they had done. ..I, the physicist had a
point. We cannot expect prospective clients to
search the optimization literature. I think it is
time to branch out from adherence to our histori-
cal roots in operations research. We are just as im-
portant to the physical applications, but we must
convince others of this. For example, I hope to see
the MPS web pages linked to a cogent guide to
optimization methods.
When the new officers took over in August 1995,
the memory of the very successful Ann Arbor
Symposium was J11 fresh in our minds, the jour-
nal had reduced its backlog to a manageable level,
the plans for the Lausanne Symposium were in ca-
pable hands, and the treasury had a comfortable
balance. Fortunately, in the few months the new
slate has been in office, not much has changed.
The Mathematical Programming Society is in
great shape.
There is one fly in the ointment. Our dues have
not increased in years, and we have reached the
point where each member costs us each year about
$20 more than the dues he/she pays. We must
raise dues. After some consideration, the Executive
Committee has decided to ask the Council to ap-
prove an increase in dues by $5 per year until we
regain equilibrium. If the Council approves this
increase, we will use the money in the treasury to
make up the difference each year that is necessary.
This decision was taken because we felt that some


of our members might find it more palatable than
a large single jump in dues. Some sister organiza-
tions avoid this situation by raising dues a bit each
year whether or not a raise is needed in a particu-
lar year. The whole issue of dues increases will be a
topic for the Council meeting in Lausanne.
So, expect some increase in dues this year. MPS
will remain a great bargain even when dues reach
the break-even point. Rest assured that there is no
move afoot to change the low-key, inexpensive,
underadministered, research-oriented character of
MPS. We do not aspire to be INFORMS, AMS,
or SIAM. These are fine organizations, but MPS
fills a different niche.
Bob Bixby (bixby@caam.rice.edu), the( 1'. of
the Publications Committee, is ( I...: of an Ad
Hoc Committee on Electronic Publishing. The
committee consists of the Editors of MPS A&B as
well as Irv Lustig, Clyde Monma, and Uwe
Zimmerman. We do not plan to lead the charge
into electronic publishing, but we do plan to posi-
tion MPS to avoid being left behind. Contact
members of the committee with your ideas.
Finally, some timely topics discussed at the recent
MPS Executive Committee meeting: we are hop-
ing for some strong proposals for the 2000 Sym-
posium; we are trying to improve our membership
services in light of persistent complaints, and OP-
TIMA has increased its pagination by adding in-
teresting interviews and feature articles.
Please contact me (dennis@caam.rice.edu) or
Executive Committee Chair Steve Wright
(wright@mcs.anl.gov) with any comments on any
society business. One of our great strengths in
MPS is that our membership is scattered across
the globe. Unfortunately, this also means that our
business meetings are very infrequent. The
internet has exciting possibilities for increased
communication among us, and we invite you to
use it to let us know what you are thinking.
Address:
John Dennis
187 CITI/Fondren MS 41
Rice University
Houston, Texas 77251-1892
Voice: (713)527-4094
Fax: (713)285-5318
dennis@caam.rice.edu
http://www.ruf.rice.edu:80/~dennis/


N? 48


PAGE 2


DECEMBER 1995





'AG 3NI-48 DEEME 1995I


continuedGrobner

bases in

integer

programming


reduced Grdbner basis with re-
spect to the cost vector c, of the
toric ideal of A. (The toric ideal
of A is a polynomial ideal con-
structed from A.) In Stage 2, we
solve IP 1'; for a b of interest
by "reducing" (with respect to
G), an arbitrary feasible solu-
tion of the program to an opti-
mal solution. A geometric inter-
pretation of the above proce-
dure, see [23], recognizes the re-
duced Gr6bner basis G, as a test
set for the family of programs
IPA,,(-). A finite set of integral
vectors T C Z" is called a test set
[18] for IPA,c(.) if, for every non-
optimal solution a to a program
in IPA,,c(), there is some vector t
e T such that a t is a solution
to the same program with a
smaller objective function value
than that of a. In practice we re-
fine the cost vector c by say the
lexicographic order on the vari-
ables so that the resulting cost
vector is a linear order on N".
This technical assumption guar-
antees finiteness of the
Buchberger algorithm, unique-
ness of the optimal solution for
each program, and a strict de-
crease in cost value from a to
a t. Clearly, if a test set for
IP(A,() is known, we have a
trivial algorithm to solve all
programs in the family, pro-
vided an initial solution is
known for each program. In
fact, the full Conti-Traverso al-
gorithm has a "Phase I" and
"Phase II" that allows one to
start with an "artificial solu-
tion" to IPA ,(b) and arrive first
at a feasible solution and finally
the optimal solution. Once the
reduced Gr6bner basis Gc is ob-
tained, Stage 2 of the Conti-
Traverso algorithm can be seen
as constructing a monotone


(with respect to c) path from the
initial non-optimal solution of
IP ,(b) to the optimal solution,
by using vectors in qc to succes-
sively move from one :. -0 -!.
solution of the program to a bet-
ter one. In effect, for every b
such that IPA,(b) is feasible, one
can use qc to build a directed
graph in P1, the convex hull of
all solutions to IPA,(b), whose
nodes are the lattice points in P1
and edges are the elements in
qG. Each such graph has a
unique sink at the optimal solu-
tion to IP,c(b).
Since test sets provide a very
natural and intuitive method
for solving an integer program,
it is not surprising that one
finds many test sets in the inte-
ger programming literature. In
1975, Jack Graver [13] showed
the existence of a finite set of
vectors that solve all programs
in the family IPA. Here IP, de-
notes all integer programs of
the form IP,c(b) but for which
both the cost and right hand
side vectors are allowed to vary.
Variants of the Graver test set
appear both in [4] and [7]. In
1981, Herbert Scarf [17] intro-
duced another test set called the
neighbors of the origin. The rela-
tionships among all these test
sets (including C r.. hi. r bases)
are discussed in [23]. In this
context, a distinctive feature of
the Gr6bner basis is that it can
be computed in practice via the
Buchberger algorithm.
Universal test sets. A set of in-
tegral vectors UA e Z" is called a
universal test set for IPA if, for
any choice of c and b, U, con-
tains a test set for IPA,(b). The
Graver test set mentioned above
is such a set. It can be shown
that every reduced Gr6bner ba-
sis 9w is contained in the Graver
test set. Since the Graver test set
is finite, it follows that there ex-
ist only finitely many distinct
Gr6bner bases associated with a
fixed matrix A as the cost func-


tion is varied. The union of
these reduced ( i..bi,. i bases,
denoted UGB,, is a minimal
universal test set for IPA c.-11. I
the universal Grdbner basis of IP .
The size of an element in UGBa
could be exponential in the size
of the data [19]. Let LP, denote
the family of linear programs
that are the linear relaxations of
programs in IPA and let Pb de-
note the feasible region of
LPA,(b). The simplex method
solves LPAc(b) by starting at a
vertex of Pb and moving mono-
tonically along edges of Pb ii 1l !
an optimal vertex is reached. It
is well known that the edge di-
rections of the polyhedra Pb, as
b varies, are precisely the mini-
mal integral dependencies of
the columns of A also known
as circuits of A. Hence the cir-
cuits of A form a universal test
set for LPA. Since we have
rational data, for every right
hand side vector b there exists a
b' such that P, is a multiple of
Pb. Hence the circuits of A are
contained in the universal
Gr6bner basis UGBA. In particu-
lar, for unimodular matrices,
the circuits constitute UGBA.
Since universal Gr6bner bases
study integer programs using
fundamentally different meth-
ods from the classical tech-
niques, they provide many new
insights into the structure of in-
teger programs. One such result
shows that the elements of
UGBA are precisely the set of all
primitive edge directions in the
polyhedra PI as b varies [21].
Hence UGBA does for integer
programs what circuits do for
linear programs. In a sense,
the Gr6bner basis method for
integer programming is the
"integer-analogue" of the sim-
plex method for linear program-
ming. Further analogies of this
nature are established in [21].


Examples and GRIN. We saw
earlier that in each polyhedron
P1 the elements of 9, build a di-
rected graph, which is a union
of directed paths from every
non-optimal lattice point in PF
to the optimal. If we disregard
the direction of edges in this
graph, then a reduced Gr6bner
basis can be thought of as pro-
viding a path between any two
feasible lattice points in Pb.
These two observations make
way for a number of applica-
tions. Most simply one can use
the first observation to enumer-
ate and count all lattice points
in PJ. This is accomplished by
first computing a reduced
Gr6bner basis q, and using this
to build a graph (as above) in
P1, whose unique sink is the
optimal solution to IP,c(b). In
order to enumerate all lattice
points in P', we reverse all
edges in this graph and search
the graph starting at the opti-
mal solution of IPA,(b) which
now becomes the root of this
graph. This '.. Ir n k.. !,," pro-
cedure was adapted in [15] to
solve a class of manufacturing
problems modeled as integer
programs with a probabilistic
side constraint given by an
oracle. In this case, it was pos-
sible to speed up computations
by exploiting the structure of
the matrices at hand.
A second application can be
found in statistics [11]. One of
the ways to check whether two
attributes of a population, for
instance, the eye color and hair
color of its members, are corre-
lated, is to construct contin-
gency tables from samples of
the population. A contingency
table in our example can be
viewed as a matrix whose ijth
entry is the number of people
from the sample with eye color i
and hair color j. We study
whether the attributes are corre-
lated by comparing the given


IIIIIIIIIIWI


--- ---


N? 48


DECEMBER 1995


PAGE 3







continuedGr6bner


bases in

integer

programming

contingency table with another
table selected at random from
the set of all tables with the
same marginal distribution
(same row and column sums).
All tables with a fixed marginal
distribution can be described as
the set of all solutions to a sys-
tem (Ax = b, x > 0, integer}.
Therefore, a Gr6bner basis asso-
ciated with A allows one to
move from one contingency
table to another with the same
marginal dlt-ril..,iti1.! This pro-
cedure can quickly generate
enough tables which ensure
that the table selected for the
comparison is close to random.
Gr6bner bases can be used to
compute primitive partition iden-
tities (ppi's) as shown in [10].
For a given n e N a ppi is any
identity of the form
a + a2 + ... + a = b + b2 ... + b,
where 0 < a,,b < n, a., b e N with
no proper subidentity
a,, +a, + ... +air = b, + 2 + ... + b,
where 1 r + s can think of ppi's as the gener-
alization of the identity 1+1=2.
It is not hard to recognize the
ppi's for a given n as the Graver
test set of the matrix A =
[1,2,...,n]. This set can be com-
puted using Buchberger's algo-
rithm. As n increases, the cardi-
nality of the set of ppi's grow
very fast; for n = 12 and n = 13
there are 9285 and 18900 ppi's
respectively. Until recently,
these could be computed effi-
ciently up to n = 13 by


MACAULAY. This has been ex-
tended up to n = 22 using a Hil-
bert basis computation [16]. The
Graver test set of the matrix A =
[(1,1),(1,2),...,(1,n)] corresponds
to the homogeneous ppi's (hppi)
which are partition identities of
the above form with the same
number of summands on the
left and right sides of both
equations. The associated ideal
is the ideal of the projective mo-
nomial curve of degree n-l, an
important curve in algebraic ge-
ometry. The champion again is
Buchberger's algorithm;
MACAULAY can compute
hppi's for n < 12. For
n = 13, there are 16968 hppi's,
and they were found by GRIN
after a lengthy computation.
GRIN (GR6bner bases for INte-
ger programming) is an experi-
mental software package which
computes Gr6bner bases of ide-
als arising from integer pro-
grams. It is intended to be a tool
for combinatorial optimization
and computational algebra and
for problems that lie in the in-
tersection of these fields. GRIN
exploits the special structure of
toric ideals, which are the ideals
that occur in this context. Due
to their special nature, these
ideals allow simple data struc-
tures and also an implementa-
tion of the Buchberger algo-
rithm that is easier, and more
efficient, than in general. A
main feature is a b 1 ilt- ii option
for computing the reduced
Gr6bner basis of a given ideal
by making "short" Gr6bner ba-
sis computations successively, a
new approach in the area. Other
state-of-the-art algorithms, such
as an algorithm due to Fausto
DiBiase and Riidiger Urbanke
[9], are also implemented. In
[14], one can find a comparison


of GRIN to existing integer pro-
gramming software (CPLEX).
The future. The connections be-
tween toric ideals and integer
programming point toward
many exciting future directions
of research. We indicate some of
them below.
An important practical issue is
the computability of C ,.. I-..
bases for integer programs. De-
vising both theoretical and pro-
gramming tricks to speed up the
Buchberger algorithm is an ob-
vious step in this direction. Like
in the classical techniques for
integer pi ...- i -, 11, computa-
tions can often be fine tuned by
exploiting the structure of the
problems. If one is interested in
solving IPAc(b) for a fixed b, then
often a small subset of the
GrBbner basis G, will suffice as
a test set. Therefore, methods
that incorporate b into the
i: .|,- -,., i. i algorithm (permit-
ting shortcuts in computations)
to produce a sufficient test set
for IPA,(b) are very worthwhile
in this respect. An idea in this
direction can be found in [24]
and [25] where the concept of a
truncated Grdbner basis is intro-
duced. This procedure produces
a test set for the family of inte-
ger programs whose right hand
side vector is "smaller than or
equal to" b in a specific sense. In
the case of 0-1 programs, trun-
cation dramatically cuts down
computational effort. Algo-
rithms that generate Gr6bner
basis elements as needed and
decide whether a given set is a
Gr6bner basis are some other
interesting issues that fall in this
general area of research.


We close by indicating some
connections between the
( t.1 -., basis method and
Ralph Gomory's group theo-
retic approach [12] to solve inte-
ger programs, which we refer to
as the group problem. (See [22]
for details about this connec-
tion.) The group problem can be
interpreted as the symmetric
analog to the usual linear relax-
ation of an integer program: in-
stead of relaxing the integrality
constraints, we remove the non-
negativity constraints on the
variables while keeping the in-
tegrality requirements. This can
be seen as a "localization" of the
ideal associated with the prob-
lem [20]. This localization leads
to an even simpler ideal, which
might provide a test set that
solves the original problem. As
in every relaxation procedure,
this method does not guarantee
to find the optimal solution at
the first attempt. Here, it would
be interesting to know whether
there are classes of integer pro-
grams for which we can guaran-
tee that the localization will
give the optimal solution imme-
diately.
The Gr6bner basis technique for
integer programming is still
very much in its infancy. Hope-
fully, the access this technique
provides to the powerful tools
of algebra and algebraic geom-
etry will help shed new light on
the structure and complexity of
integer programs.


II^


PAGE 4


N 48


DECEMBER 1995





PAGE 5 IN" '10

I -


References
[1] W.W. Adams and P.
Loustaunau, An Introduction to
Grdbner Bases, American Math-
ematical Society, Graduate Studies
in Math., Vol. III, 1994.
[2] D. Bayer and I. Morrison (per-
sonal communication).
[3] D. Bayer and M. Stillman,
MACAULAY: A computer algebra
system for algebraic geometry.
Available by anonymous ftp from
zariski.harvard.edu.
[41 C.E. Blair and R.G. Jeroslow,
The value function of an integer
program, Mathematical .. ."'
.,., 23(1982) 237-273.
[5] B. Buchberger, Gr6bner bases -
an algorithmic method in polyno-
mial ideal theory, Chapter 6 in
N.K.Bose (ed.): Multidimensional
Systems T' . D. Riedel Publica-
tions 1985.
[6] P. Conti and C. Traverso,
Gr6bner bases and integer pro-
gramming, ; .'... .' AAECC-9
(New Orleans), Springer Verlag,
LNCS 539 (1991) 130-139.
[7] W. Cook, A.M.H. Gerards, A.
Schrijver and t. Tardos, =. ,,. -i i
theorems in integer linear pro-
gramming, Mathematical 7' ..
34 (1986) 251-264.
[8] D. Cox, J. Little and D. O'Shea,
Ideals, Varieties and. \I .''. An
Introduction to Computational Alge-
braic Geometry and Commutative Al-
.I ..: Undergraduate Texts in
Math, Springer Verlag 1992.
[9] F. DiBiase and R. Urbanke, An
algorithm to compute the kernel of
certain polynomial ring homomor-
phisms, Experimental Mathematics,
to appear.
[10] P. Diaconis, R. Graham and B.
Sturmfels, Primitive partition
identities, in:Paul Erdds is 80. Vol.
II, Janos Bolyai Society, Budapest
(1995), 1-20.
[11] P. Diaconis and B. Sturmfels,
Algebraic algorithms for sampling
from conditional distributions, An-
nals of Statistics, to appear.
[12] R.E. Gomory, Some polyhedra
related to combinatorial problems,
Linear Algebra and its, lTi .1.. -.. 2
(1969) 451-558.


[131 J.E. Graver, On the founda-
tions of linear and integer pro-
gramming I, Mathematical '... .;,*
ming 8 (1975) 207-226.
[14] S. Hosten and B. Sturmfels,
GRIN: An implementation of
Gr6bner bases for integer pro-
gramming, in: Integer ''.... .
and Combinatorial Optimization, E.
Balas, J. Clausen (eds.), Springer
Verlag, LNCS 920 (1995) 267-276.
[15] S.R. Tayur, R.R. Thomas and
N.R. Natraj, An algebraic geom-
etry algorithm for scheduling in
presence of setups and correlated
demands, Mathematical"' .. .
ming 69 (1995) 369- 401.
[16] D.V. Pasechnik, RIACA/CAN
(personal communication).
[17] H.E. Scarf, Neighborhood sys-
tems for production sets with
indivisibilities, Econometrica 54
(1986) 507-532.
[18] A. Schrijver, Theory of linear
and :..j. i ... ... ... ... W iley-
Interscience Series in Discrete
Mathematics and Optimization,
New York (1986).
[19] B. Sturmfels, GrBbner bases of
toric varieties, T6hoku Math. Jour-
nal 43 (1991) 249-261.
[20] B. Sturmfels, Grdbner bases and
convex polytopes, American Math-
ematical Society (1996), to appear.
[21] B. Sturmfels and R.R. Thomas,
Variation of cost functions in inte-
ger programming (1994), Techni-
cal Report, School of Operations
Research, Cornell University.
[22] B. Sturmfels, R. Weismantel
and G. Ziegler, Gr6bner bases, lat-
tice polytopes and corner polyhe-
dra, i. .. ... zur Geometrie und Al-
gebra, CG.l,.:.!.t..-.,. to Algebra and
Geometry 36 (1995), No.2, 281-298.
12 R.R. Thomas, A geometric
Buchberger algorithm for integer
programming, Mathematics of Op-
erations Research, to appear.
[24] R.R. Thomas and R.
Weismantel, Truncated Gr6bner
bases for integer programming
(1995), preprint SC 94-29,
ZIB Berlin.
[25] R. Urbaniak, R. Weismantel
and G. Ziegler, A variant of
Buchberger's algorithm for integer
programming I !' 4,. preprint SC
94-29, ZIB Berlin.


DECEMBER 1995
S


In this issue we feature a versatile
and robust nonlinear .. -, .,.,, -,< ..
software package that has been avail-
able to the academic and research
community as a Beta release for sev-
eral years. While a detailed user's
guide has also been available, this E
guide will be integrated into a SIAM
book that also includes theory
and real world applications. C O I'" T '1 l .
The:..II. ,.. article has


been adapted from the first few pages
of the User's Guide and from corre-
spondence with the authors.


FSQP: A Versatile Tool for
Nonlinear Programming User's
Guide, by C.T. Lawrence, A.L. Tits,
i, ,..| 1 .. i t 1996(forth-
coming). This book is intended as a de-
tailed user's guide to the two feasible
sequential quadratic programming
(FSQP) packages, CFSQP (C version)
and FFSQP (FORTRAN version). It
also presents an in-depth explanation of
the FSQP algorithm and includes ex-
amples of diverse applications from sci-
ence and engineering. The algorithm
can be used to solve three families of
problems: (P1) find a solution to a sys-
tem of nonlinear equations (feasibility
problem); (P2) find a solution to a sys-
tem of nonlinear constraints I-..'.,
equality and inequality) that optimizes
a given function (standard nonlinear
programming problem); and (P3) find
a solution to a system of nonlinear con-
straints that minimizes the maximum
(or maximizes the minimum) of a set of
functions (constrained minimax (or
maximin) problem). The functions are
.. r. , .. i differen-
tiable; however, it frequentlyworks well
when this assumption is violated. More-
over, special techniques are employed to
exploit the structure oflinear constraints
for added efficiency.
TheMethod.Sequential quadratic pro-
gramming (SQP) is a superlinearly con-
vergent algorithm for finding approxi-
mate local solutions to nonlinear pro-
gramming problems (see P.E. Gill, W.
Murray, and M.' ,1- Practical Op-
timization, Academic Press, New York,
i ':i I . ,i .! ; .


obtained can be locally suboptimal or
superoptimal and feasible or infeasible,
depending on the functions in the prob-
lem and the feasibility and optimality
tolerances used. ;1.;T I, i ..,' ..
anteed in the limit. This approach has
enjoyed wide success on applications
whose constraints are not rigid. In many
importantapplications, however, violat-
ing some or all of the constraints is not
acceptable. This is the case when the
objective function is not defined outside
the feasible set. For example, dynamical
systems must be stable in order for cer-
tain steady state errors to be well-de-
fined. Another reason for generating fea-
sible iterates is the requirement in some
applications (such as certain real-time
control problems) that certain "hard"
constraints must be satisfied after a pre-
scribed amount of time.
Another situation where feasibility of
successive iterates is imperative for some
constraints arises in the interactive pro-
cess used for designing engineering sys-
tems (see W.T. Nye and A.L. Tits, "An
Application-Oriented, Optimization-
Based Methodology for Interactive
D e ;. i. r. .T ,.;,. B c ,..,,, ."Inter-
national Journal of Control 43 (1986)
1693-1721). In such problems, some of
the specifications can be relaxed, but
others (such as stability or physical
realizability) cannot. There are usually
tradeoffs for violating the "soft" con-
straints (specifications) which can be
explored by the designer during the
design process. This tradeoff explora-
tion, however, is only meaningful after
the "hard" constraints iti. ,'-..,
are satisfied. Since each iteration of an
c. r.; ... i I:. .;d... ;., olves one or
more function evaluations, and since in
a typical design environment function
evaluations call for ... ,f ., .. 'p ..... II
: ., -


--






N048I~ DECMBE 199


sofiVr&tere cornputodlion


, 11 required that hard constraints be
satisfied at each iteration.
First order methods of feasible directions
have been proposed since the late 1950s
byZoutendijk, Polak, Pshenichnyi, and
i I I 1 ,, I,,, 1
linear rates of convergence. Motivated
by the need for feasible iterates in large
applications, a group ofresearchers at the
University of Maryland set out to
modify the SQP method to generate
feasible iterates without sacrificing its
superlinear rate of convergence. The
algorithm's main architect was Dr.
E liane R .:' I I. I I..... i,. 11 il.
University of Maryland. Development
of the methodology and its implemen-
tations has been ongoing at the
University's Institute for Systems Re-
search. The key feature that distin-
guishes FSQP from other SQP algo-
rithms is the concept ofa "semi-feasible"
point. In solving nonlinear program-
ming (P2) and constrained minimax
([ I.,l~l., [,. [ in I ',' I.l ,, ,,,, ,
determines a point that satisfies all in-
equality constraints and linear equality
-. 1, ,,, I J I , .I ;. r, . II.. th e
nonlinear equality constraints. Such a
point is called semi-feasible. The algo-
rithm subsequently generates sequence
of semi-feasible points while striving to

and to optimize the objective function.
On the other hand, FSQP makes use of
a classical variable metric scheme to
estimate the Hessian of the Lagrangian.
Asa consequence, it is not well-suited for
problems that involve a "very large"
number of decision variables.
The core of the FSQP algorithm only
deals with inequality constraints (and
linear equality constraints). Given a
point x satisfying the constraints, the
basic SQP search direction da may not
be a feasible direction; i.e., even short
steps along this direction may yield
points that do not satisfy the constraints.
Yet dis at worst "tangent" to the feasible
set X. Thus, in FSQP, dO is slightly
"tilted" toward the interior ofX, to yield
the search direction d. The amount of
tilting is closely monitored in order to
preserve the quasi-Newton convergence
properties of the SQP direction.


A further adjustment is needed in order
to prevent a Maratos-like effect. In a
nutshell, the Maratos effect stems from
the near conflict between the need to
possibly reduce the step length along the
search direction, to prevent II ,
or divergence in the early iterations, and
the need to allow a full unit step when
a solution is approached, to allow fast
convergence to take place. In the context
ofFSQP, reduction ofthestep length in
early iterations is necessary not only to
avoid divergence, but also to ensure
feasibility of the successive iterates. Two
schemes are available in FSQP to ensure
that a full unit step will always be ac-
cepted close to a solution: (i) a second
order correction, with I.. 1., '" of the
search direction, and (ii) a
nonmonotone line search. On the aver-
age, the latter .11. -i ... 1 i II faster
progress towards the solution as it often
yields a larger step size. On the other
hand, the former has the property that
the value of the objective function im-
proves at each iteration, which can be an
important advantage in certain applica-
tions.
Finally, nonlinear equality constraints
are dealt with by means of a technique
first suggested by Mayne and Polak (see
D.Q. Mayne and E. Polak, "Feasible
Direction Algorithms for Optimization
Problems with Equality and Inequality
Constraints," Mathematical Program-
ming 11 (1976) 67-80) in the context
of first order methods of feasible direc-
tions. The idea is as :.. 1I., split each
equality constraint into two inequalities
("<" and "2"). Include one of these (the
one satisfied by the initial iterate, say,
h(x) < 0) with the other nonlinear in-
equality constraints and penalize viola-
tions ofthe other one (h(x) > 0) by means
ofa simple penalty function: a multiple
-ch(x) (for c>0) of the violation is added
to the objective function (or to the
maximum among the objective func-
tions). The advantages of this scheme
over the more common quadratic pen-
alty function are that (i) convergence to
a feasible point is achieved without driv-
ing cto infinity (i.e., this is an exactdif-
ferentiable penalty function) and (ii) one
side of the constraints (here h(x) < 0) is
satisfied throughout the optimization
process ("semi-feasibility"), which is a
desirable feature in certain applications.


The Software: I .. n r .. ..ofthe
FORTRAN implementation (by the
third author of this book) was released
in 1989. The first C implementation (by
the first author of this book) was released
in 1993. The successive versions of both
packages were made widely available on
the Internet. As a result, the authors re-
ceived voluminous :. .l -.I over the
years from the user community. This led
to elimination of bugs and improved
implementations and enhancements.
The C version (CFSQP) includes a spe-
cial scheme to efficiently handle prob-
lems with a large number ofeither (mini-
max) objective functions or inequality
constraints relative to the number of
decision variables. Such problems often
arise in engineering applications.
CFSQP is especially tuned for the case
where groups of such constraints or ob-
jective functions are identified by the
userasbeing" 1n.i 'I1 ill,; ..I i...I,"i.e.,
I I_,, l..... ', r "offu actions
where each one is nearly identical to its
predecessor and to its successor. This is
the case, for instance, with groups of
constraints arising from the
discretization of a continuum of con-
straints (semi-infinite optimization).
Intuitively, if dis a direction of descent
for one constraint in a "sequentially
related" list, it is also a descent direction
for nearby constraints in the list. The
CFSQP implementation of FSQP ex-
ploits this observation by computing
successive search directions based on a
...,111.. lg,,I ,_,,,4 ,i, r ..r h,...,I.j..-.
tives/constraints, with an ensuing re-
duced computing cost per iteration and
a decreased risk of numerical difficulties.
This subset is updated at each iteration
in such a way that global convergence is
ensured. This scheme ,i ....... ,I II ac-
celerates execution times of problems
with "sequentially related" constraints
over the FORTRAN version.
Both FFSQP and CFSQP run on just
about any platform. The authors incor-
porate modifications into new releases
whenever they are notified of system
compatibility problems. The software
has been designed to be a valuable re-
search and development tool and has
enjoyed a good reputation as a reliable
and versatile engine that can be inte-
grated into other packages. One user has
interfaced it to MATLAB and is report-
ing good results. Another user is devel-
oping an interface to Octave.


_ ,, I. ,-, T ;, i,, 1 1; 1 . r ,l .l 1;. 1 ;..1 ,,
that have used the package with success:
1. Minimizing reconstruction noise
in Magnetic Resonance Imaging
(MRI). Uses "sequentially related"
constraints feature of CFSQP to re-
duce computational effort.
2. Obtaining the "best" description of
clutter noise in over-the-horizon ra-
dar. The feasibility feature of FSQP is
required for some of the models.
3. Dynamic Manipulation. Robotic
manipulation planners that exploit
dynamic effects rather than ignoring
them or attempting to cancel them
out.
4. Optimization-based design ofhub-
and-shaft assemblies for dual-wheel
excavators.
5. Optimal Protein Separation. Using
ionic strength as a control variable, a
piece-wise constant optimal control
problem is solved as a sequence of op-
timal parameter selection problems.
6. Parametric Surface Polygonization.
A polygonal mesh representative of a
surface is constructed.
7. I i... .. : the dose effect for the
analysis of intermediately lethal tu-
mors," A. J. Rossini and L. Ryan,
preprint, Pennsylvania State College
of Medicine, Hershey, PA 17033.
8. "Synthesis of hierarchical traffic
control systems," Ludmil Mikhailov,
Technical report, Universite Libre de
i'., II. Laboratoire
d'Automatique, February 1993.
9. "Design of redundancy relations
for failure detection and isolation by
constrained optimization," Michel
Kinnaert, preprint, Universite libre
de Bruxelles, Laboratoire
d'Automatique, July 1992.
10. Screening multi-purpose reservoir
systems optimization model for siz-
ing and selecting among several po-
tential reservoir sites.
If you are interested in more informa-
tion about the software or the appli-
cations, please contact:
Prof. Andrd L. Tits, Dept. of Electri-
cal Engineering and Institute for
Systems Research, University of
Maryland, ( II .. Park, MD 20742,
USA
Phone (301) 405-3669
Fax (301) 405-6707
E-mail: andre@eng.umd.edu
-FAIZ AL-KHAYYAL


-~---~


N? 48


DECEMBER 1995





'\Gf 7 NT~l48 DllcEMBER 19951


.\












How to access information about

Mathematical Programming

Information about Mathematical Programming, such as
table of contents, policy, and subscription can be found
via the homepage o(t 1i ..: Science Publishers.
Follow these instructions:
1. Open URL: http://www.elsevier.nl/
2. Click on "ESTOC (Elsevier Science Table of Contents)"
3. Click on "Alphabetical listing for all fields"
4. Click on "M"
5. Click on "MATHEMATICAL PROGRAMMING"
Voila!
-GERARD WANROOY


Volume 70, No. 1

A. Prekopa and W. Li, "Solution
of and bounding in a linearly
constrained optimization problem
with convex, polyhedral objective
function."
N. Garg and V.V. Vazirani, "A
polyhedron with all s-t cuts as
vertices, and adjacency of cuts."
K.G. Murty and S.-J. Chung,
..;. i ., t -, in enumerating faces."
J.E. Falk and J. Liu, "On bilevel
programming, Part I: General
nonlinear cases."
R. Schultz, "On structure and
stability in stochastic programs
with random I lii, .i. ;,i matrix
and complete integer recourse."
A.L. Dontchev, "Implicit function
theorems for generalized equa-

G7 Zhao and J. Zhu, "The
uI,., ,L int, ,hi.l and the
W ,,l114 l'i 1 -t 1 ,'( I ear
,iri'lli IaIt, I ii problems."


Volume -I. No. 2

J.F. Bonnns and A. Sulem,
"Pseudo power expansion of
solution of generalized equations
and co trained optimization
problems."
\ lhaipiro "Directional differen-
ti hbilitu ,ir the optimal value
li' tiui 1, i l, 'onvex semi-infinite
I'l,, --1i,010111i ."
11. Ralpl .ind S. Dempe, "Direc-
tional derivatives of the solution
of a parametric nonlinear
program."
J.A. Hoogeveen and S.L. van de
Velde, "Stronger Lagrangian
bounds by use of slack variables:
Applications to machine schedul-
ing problems."
S. Schaible and J.-C. Yao, "On the
equivalence of nonlinear
complementarity problems and
least-element problems."


A. Frank and Z. Szigeti, "A note
on packing paths in planar
graphs."
C.A. Hane, C. Barnhart, E.L.
Johnson, R.E. Marsten, G.L.
Nemhauser and G. Sigismondi,
"The fleet assignment problem:
solving a large-scale integer
program."


Volume 70, No. 3

J.P. Dussault and Y. Gningue,
"Unification of basic and
composite nondifferentiable
optimization."
P.E. Gill, W. Murray, D.B.
Poncele6n and M.A. Saunders,
"Primal-dual methods for linear
programming."
J. Renegar, "Lii, ia I" ,1,m ,Iilil..
complexity theory and elementary
functional analysis."


Volume 71, No. 1

Y.T. Ikebe and A. Tamura, "Ideal
polytopes and face structures of
some combinatorial optimization
problems."
S. Kim, K.-N. Chang and J.-Y.
Lee, "A descent method with
linear programming subproblems
,-., iiiliff, I, iii,rl'l convex
optimization."
M. Laurent and S. Poljak, "One-
third-integrality in the max-cut
problem."
C. Chen and O.L. Mangasarian,
"Smoothing methods for convex
inequalities and linear
complementarity problems."
J. Brimberg, "The Fermat-Weber
location problem revisited."
A. Auslender and M. Haddou,
"An interior-proximal method for
convex linearly constrained
problems and its extension to
variational inequalities."
E. Carrizosa and F. Plastria, "On
minquantile and maxcovering
optimisation."


N? 48


DECEMBER 1995


PAGE 7






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Eii E t-, G L Nemhause A H G Rinnooy Kan ;-c M J To.'d
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Fo -ve fate sevc us e-mal fa or teehn an quot the foloin nube wit al Srders/requests 91.9071






I'ACE N0 48DECEMBR 199


Conference


P Third Workshop on Global
Optimization, Szeged, Hun-
gary, Dec. 10-14, 1995
1 Conference on Network
Optimization, University of
Florida, Feb. 12-14, 1996
Workshop on
SATISFIABILITY PROBLEM:
THEORY AND APPLICATIONS
Rutgers University
March 11-13, 1996
5th SIAM Conference on
Optimization, British Columbia,
May 20-22, 1996.
I 18th Symposium on Math-
ematical Programming with
Data Perturbations,
George Washington University,
23-24 May 1996.
) IPCO V, Vancouver, British
Columbia, Canada,
June 3-5, 1996
Fifth International Sympo-
sium on Generalized Convexity
Luminy-Marseille, France
June 17-21, 1996
) IFORS 96 14th Triennial
Conference, Vancouver,
British Columbia, Canada,
July 8-12, 1996
IRREGULAR 96
Santa Barbara, California
Aug. 19-23, 1996
International Conference on
Nonlinear Programming,
Beijing, China,
Sept. 2-5, 1996
XVI International
Symposium on Mathematical
Programming, Lausanne,
Switzerland, Aug. 1997


,!o re n Ie-,


The EIGHTEENTH Sym- C a
posium on Mathemati-
cal Programming with
Data Perturbations will
be held at George
Washington
University's Marvin Center on 23-
24 May 1996. This symposium is
designed to bring together practi-
tioners who use mathematical pro-
gramming optimization models
and deal with questions of sensi-
tivity analysis, with researchers
who are developing techniques
applicable to these problems.
Contributed papers in mathematical
programming are solicited in the fol-
lowing areas:
1. Sensitivity and stability analysis re-
sults and their applications.
2. Solution methods for problems in-
volving implicitly defined problem
functions.
3. Solution methods for problems in-
volving deterministic or stochastic pa-
rameter changes.
4. Solution approximation techniques
and error analysis.


I


F


Irregular Workshop
The series of workshops on .' .' I -
rithms for Irregularly Structured Problems
- IRREGULAR'9X started in Geneva in
August 1994 and was held in Lyon in Sep-
tember 1995. These workshops address is-
. 1 I r .1 1 J. .., L u_ 1.H 1 1. .. ,l .I ..
lutions to irregularly structured problems.
In fact, efficient parallel solutions have been
found to many problems. Some of these
solutions can be obtained ,,r..... ,i.. 11
from sequential programs using compilers.
However, there still exists a large class of
problems, known as irregular problems, that
lack efficient solutions.
The aim of the IRREGULAR series is to
foster cooperation among practitioners and
theoreticians in the field. It covers such
topics as approximated and randomized
methods, automatic synthesis, branch and
bound, combinatorial optimization, com-
;il.i,. computer vision, load balancing,


tons that describe
problems in sensitivity
B or stability analysis en-
countered in applica-
tions are also invited.
Abstracts of papers intended for pre-
sentation at the Symposium should be
sent in triplicate to Professor Anthony
V. Fiacco. Abstracts should provide a
good technical summary of key results,
avoid the use of mathematical symbols
and references, not exceed 500 words,
and include a title and the name and
the full mailing address of each author.
The deadline for submission of ab-
stracts is 17 March 1996.
Approximately 30 minutes will be
allocated for the presentation of each
paper. A blackboard and overhead pro-
jector will be available.
ANTHONY V. FIACCO, Organizer
Sponsored by the Department of Op-
erations Research and the Institute for
Management Science and Engineering
..- h .I..l...Fr ,,:_ ,....i .1;.. [Science
The George Washington University
Washington, D.C. 20052
Telephone: (202) 994-7511


parallel data structures, scheduling and
mapping, and sparse matrix and symbolic
computation.
The papers presented in Geneva were pub-
lished in a book by Kluwer Academic Pub-
lishers. In 1995, the proceedings were pub-
lished in LectureNotes in Computer Sciences
(LNCS) by Springer-Verlag.
IRREGULAR'96 will take place in Santa
Barbara, California, from August 19-23,
1996. Its proceedings will be published
again in LN( '. ., I il be available at the
workshop.
For the call for papers and further in-
formation, please contact one of the
IRREGULAR co-chairs:
Afonso Ferreira
ferreira@lip.ens-lyon.fr
or Jose Rolim
rolim@cui.unige.ch
-PANOS PARDALOS


--~---~


DECEMBER 1995


N? 48


PAGE 9


-i-

.r
3;
-i
'-'- r
..--ir


I ~






PAG 10 N48 DECEMBER 1995lm


First Announcement
Fifth International Symposium
on Generalized Convexity
Luminy-Marseille, France
June 17-21, 1996
After the NATO Advanced Study Insti-
tute on Generalized Concavity in Optimi-
zation and Economics in Vancouver,
Canada (1980), and similar symposia in
Canton,NY(19 1 !I .1(1988)and
PFcs, Hungary (1992), we are glad to an-
nounce Generalized Convexity 5. This
symposium will be held at Centre Inter-
national de Rencontres Math6matiques
(CIRM), Luminy, near the Mediterra-
nean Sea. In addition to CIRM, sponsors
are the Mathematical Programming So-
ciety (MPS) and the recently founded
Working Group on Generalized Convex-
ity (WGC) within MPS.
This symposium attempts to solve open
problems related to theoretical, algorith-
mic, computational and modeling issues
in connection with generalizations of
convexity, as they arise in mathematical
programming, economics, management
science, engineering, applied sciences, nu-
merical mathematics, etc. An emphasis
will be placed on the discussion of gener-
alized monotonicity, a new area of re-
search in the '90s, relevant to variational
inequalities and equilibrium problems.
F t... I -, ..,, ,i1 l ,,,- ,. .
to relate generalized convexity more
closely to neighboring fields such as
nonsmooth analysis, economic theory,
complementarity theory/variational in-
equalities and stochastic programming.
The ..II.. .. scholars have tentatively
agreed to participate through tutorials in
these areas: F. Clarke, A. Mas-Colell, J.S.
I and R. Wets.
The International Scientific Committee
ofWGC serves as Program Committee
and consists of S. Schaible, USA (chair),
C.R. Bector, Canada, B.D. Craven, Aus-
tralia, J.-P. Crouzeix, France, J.B.G.
Frenk, The Netherlands, S. Komlosi,
Hungary,J.E. Martinez-Legaz, Spain, and
P. Mazzoleni, Italy.
A Second Announcement will be sent to
those who preregister. If possible, use e-
mail. Please contact the chair of the Or-
ganizing Committee:
Jean-Pierre Crouzeix (gcv5), Applied
Mathematics, Universitd Blaise Pascal,
E-63177 Aubiere Cedex, FRANCE
E-mail:
crouzeix@ucfma.univ-bpclermont.fr
FAX: +33 (73) 49 70 64
Phone: +33 (73) 40 70 54


First International Conference on Complementarity Problems
Johns Hopkins University
Baltimore, MD


The first international conference on complementarity problems
was held at the Johns Hopkins University in Baltimore from
November 1-4, 1995. The meeting was attended by over 50 re-
searchers from around the world, including attendees from Aus-
t, I I ...... Canada, theCzechRepublic, Denmark, Finland,
Germany, Great Britain, India, Italy, Japan, The Netherlands,
Norway, .., r... Sweden and the United States. The meeting
S .... .I ;1, i.r,.. I ..... l.,;.,;, ,.. together researchers ofthe
,. ,1, ,,,,, ; .1 ,- .. ., ,,,,,,,;, ... .. .. i- .. ,,', .. . .
and experts in a variety of applications areas. The purpose of the
meeting was to create increased cross fertilization and communi-
cation between these areas. In particular, a better understanding
and appreciation of the different aspects that each of these areas
considers is expected to be beneficial for the effective solution of
practical complementarity problems arising from applied disci-
plines. As such, we believe that the meeting was a great success.
With more than three decades of research, the subject of
complementarity problems has become a well-established and
: ,, ,r t l h i ,,, .1, ,, ,rl, .,,, ,,r, ', ,,,',,,,,,,:_ ,I,, F ,
complementarity problems are diverse and include many prob-
lems in engineering, economics, and the sciences. Several mono-
graphs and surveys have documented the basic theory, algorithms,
and applications of complementarity problems and their role in
optimization theory.
The meeting started with an overview of the complementarity field
atwhichstage a newweb site, CPNET, http://www.cs.wisc.edu/
cpnet/ was announced. When completed, this site will eventually
contain up-to-date information on upcoming conferences in the
area, a list ofactive researchers and pointers to work on algorithms,
applications and software.
Currently, the page contains a list of all the researchers present at
the conference, along with papers and software that outline some
developments in the area. T1l; ... i ,. .... ,,, i i- ...
test problems for MCP, MCPLIB, and the COMPLE-
MENTARITYTOOLBOX, suite of programs and routines for
use in conjunction with MATLAB. It is hoped that CPNET will
1II.. -- I. I r ... to grow considerably to include many new
I.I i .i. 11,, .l. ,,; 1 -...1.1 n, There are som e pointers to

formulated in standard modeling languages.
There were various themes that developed during the meeting.
Several speakers introduced new extensions ofthe basic framework
and cited applications that needed such extensions. Some new
theoretical results were outlined relating to vertical, horizontal and
extended linear complementarity problems, along with several
ideas to unify these areas. Other speakers considered noncoopera-
tive and stochastic game theory and outlined existence results and
algorithms for their solution. Variational and bimatrix inequali-
ties also drew the attention of several talks. Merit functions and
..... l-, ;,, techniques were also popular topics.
One extension that received considerable attentionwas the Math-
ematical _'.:.. .... with Equilibrium Constraints (MPEC). Sev-
eral algorithms were given for the solution of these problems, and
lots of discussion resulted during application talks relating to
reformulating problems into this framework. This appears to be
a very fruitful area for future research.


Contact problems are a rich source ofcomplementarity problems.
For these problems, complementarity is the result of the contact
condition which stipulates that the gap between two objects in
contact is either zero, or the pressure between them is zero. Clas-
.. .. .1 I ,, .1.1 ,, ,, ,. I. . I..] , 1, 1 .. I. -
vection and diffusion. An interesting use of complementarity in
contact mechanics arises in robot design, and key features of the
problem that can be modeled in the new framework include slid-
ing, friction and rigid body properties. Structural mechanics has
also used complementarity models in studies of material elasticity

up these areas to the field in general.
Complementarity has been used in economics for along time. The
renowned Walrasian law of supply and demand in general equi-
librium theory states that either there is excess supply or the price
of the corresponding good is zero. Several extensions of this basic
idea were outlined in 11 I .1 I ', with oligopolistic equilibria,
integrated assessment for problems in energymodeling, relocation
effects due to the European Common Market and the National
Energy Modeling System (NEMS). The use of similar models for
, ,t , ,,_ ,,,,,, ,, I . . , ;, i,, l ... . . .. i I ,

gestion analysis were presented at the meeting.
Several new algorithmic developments were outlined. Some of
these involved the traditional simplicial and pivotal based tech-
niques while others used novel reformulations of the
complementarity problem both as smooth and nonsmooth sys-
tems of nonlinear equations. A very popular approach takes sys-
tems of nonsmooth equations and applies a smoothing so that
traditional Newton based techniques could be applied. Still other
methods were based on quadratic programming and proximal
points formulations. New computational extensions were also
outlined. Several talks introduced new merit functions that will
prove useful in error analysis and future algorithmic design.
In conclusion, the meeting showed that the field of
complementarity research is a burgeoning area. There are already
many interesting algorithms for solving complementarity prob-
lems, along with fairly sophisticated techniques for analysis and
computation. T'. i, ;, ,i,...,,,, erofnewapplicationareas
that use this framework will require even more sophisticated
solution techniques. Furthermore, it is clear that even more ap-
plications will be developed that use complementarity modeling
in some form or other,:, ,..,'.. ... portion of which was made
possible by this meeting.
A refereed proceedings of this meetingwill be published in 1996
by SIAM. Further developments in this area will undoubtedly be
reported at the next International Conference on
Complementarity Problems. Planning is already underway, and
the conference is tentatively set for July 1998 to be held in Madi-
son, Wisconsin.
Michael Ferris
Computer Sciences Department,
University of Wisconsin, Madison
ferris@cs.wisc.edu
Jong-Shi Pang
Department of Mathematical Sciences
The Johns Hopkins University, Baltimore
jsp@vicp.mts.jhu.edu


lb-


N? 48


PAGE 10


DECEMBER 1995









ismp 7


International Symposium on Mathematical Programming
Lausanne, Switzerland, August 24-29, 1997


First Announcement

The International Symposium on Mathematical Programming is the triennal scientific meeting of the Mathematical
Programming Society. The 16th Symposium will be held at the Swiss Federal Institute ofTechnology in Lausanne, Aug.
24-29, 1997, the year of the 50th anniversary of George Dantzig's Simplex Method for Linear P .. -,i i.-in_.


O, ,,i: in, Committee: Chair: Th.M. Liebling. D. de Werra, K. Frauendorfer, K. Fukuda, H. Gr6flin, A. Haurie, A. Hertz, P. Kall, J. Kohlas, B. Lara,
H.-J. Lithi, D. Naddef, P. Neusch, F.-L. Perret, A. Prodon, P. Stahly, J.-P. Vial, M. Widmer. D. Miller (Head coordinator). InternationalAdvisory
Committee: Chair: D. de Werra. R. Ahuja, M. Akgul, K. Al-Sultan, E. /.! I.. i, K.M. Anstreicher, J. Araoz, M. Avriel, A. Auslender, A. Bachem,
E. Balas, M. Balinski, A. Ben-tal, D.P. I.. rr. .1-, C. Berge, R. E. Bixby, P. Bod, A. Buckley, R. E. Burkard, V. Chandru, S.Y. ( I .. S.J. Ci n V.
Chvatal, A. R. Conn, R. Correa, G.B. Dantzig, M.A.H. Dempster, J.E. Dennis Jr., L. C.W. Dixon, J. Dupacova, B. C. Eaves, Y. Ermoliev, S.C. Fang,
F it. 1. .., Frank, S. Fujishige, S. Gass, F. Giannessi, Ph. Gill, J.-L. Goffin, D. Goldfarb, C.C. Gonzaga, N. I.M. Gould, R.L. Graham, M. Gr6tschel,
H.W. Hamacher, P.L. Hammer, A.J. Hoffman, K.L. Hoffman, M. Iri, A.N. lusem, E. L. Johnson, J. Judice, S.N. Kabadi, R. Kannan, N. Karmarkar,
R.M. Karp, A.V. Karzanov, L. Khachiyan, V. !, 1.. '. I .... jima, H. Konno, B. Korte,J. Krarup, H.W. Kuhn, C. Lemarechal,J. K. Lenstra, P.- Ti I i...i ..I .
F. Louveaux, L. LovAsz, Th.M. I ;. 1 H1;i F.M.'.1 .I!,.!.' T.L. Magnanti, S. Maya, F. McDonald, N. Megiddo, K.' 1. I I I...,, G. Mitra, S. Mizuno, S.R.
Mohan, B. Murtagh, G.L. Nemhauser,J. Nocedal, M.W.I i -. i i .-S. Pang, K. Paparizos, P. Pardalos, C. Perin, B. Polyak, M.J.D. Powell, A. Prekopa,
W. R. Pulleyblank, L. Qi, M.R. Rao, A.H.G. Rinnooy Kan, R.T. .. 1, 1. I l I, J.B. Rosen, H.E. Scarf, R.B. Schnabel, A. Schrijver, N.Z. Shor, J. Stoer,
E. Tardos, J. Tind, M.J. Todd, Ph. L.M.J. Toint, P. Toth, A. Tucker, H. Tuy, S. W. Wallace, A. Weintraub, R.J-B. Wets, H.P. Williams, P. Wolfe, L.A.
Wolsey, M.H. Wright, S. Wright, Y. Ye, M.Y. Yue, J. Zow( iit,, -ii i 1 loisory Committee:Chair: B. Korte. J.R. Birge, C.C. Gonzaga, A. Schrijver.


Tentative list of topics
Sessions on the following topics are planned.
Si _.. r; ... for further areas to be included are welcome.
1. Linear, integer, mixed-integer programming
2. Interior-point and ip il.l i.. I... algorithms
3. Nonlinear, nonconvex, nondifferentiable, global optimization
4. Complementarity and fixed point theory
5. Dynamic and stochastic programming, optimal control
6. Real-time optimization
7. Game theory and multicriterion optimization
8. Combinatorial optimization, graphs and networks, matroids
9. Computational complexity, performance guarantees and
quantum computation
10. Approximation methods, heuristics
11. Local search, simulated annealing, tabu search, etc.
12. Computational geometry, VLSI-design
13. Computational biology
14. Implementation and evaluation of algorithms and software
15. Large-scale mathematical programming
16. Parallel computing in mathematical programming
17. Expert, interactive and decision support systems, neural
networks, fuzzy logic
18. Simulation, optimization in discrete event simulation
19. Mathematical programming on personal computers
20. Teaching in mathematical programming
21. Applications of mathematical programming in industry, government,
economics, management, finance, transportation, engineering, energy,
environment, agriculture, sciences and humanities

Site
The Symposium I take place at the Swiss Federal Institute of Technology
(EPFL) in Lausanne, Switzerland. Hotels of various categories as well as low
price accommodations will be available.
Preregistration
It is necessary to preregister in order to be included in our mailing list. This
procedure is free of charge. The second announcement, due to appear in 1996,
will be sent only to those who preregister.
(See reverse side)


E-mail address of ISMP 97
ismp97@masg1 .epfl.ch
WWW server
http://dmawww.epfl.ch/roso.mosaic/ismp97/welcome.html
(We strongly encourage you to use it).
Mailing address
ISMP 97
c/o Prof. Thomas M. Liebling
EPFL-DMA
CH-1015 Lausanne (Switzerland)
phone: + 41 21 693 2595
fax: + 41 21 693 4250
Dates & deadlines
Aug. 31, 1996: Preregistration
Sept. 1, 1996: Second announcement
April 30, 1997: Submission of titles and abstracts and early registration
Aug. 24-29, 1997: The Symposium
Call for papers
Papers on all theoretical, computational and practical aspects of mathematical
programming are welcome. The presentation of very recent results is encour-
aged. All abstracts will be available via WWW.
Structure of the meeting
A large number of people will be invited to organize sessions. Other people
may, from their own initiative, propose themselves as session organizers. If
they do so, they must write us and once our program committee has agreed,
I. .11 be included in the list. During the plenary opening session, the fol-
lowing prizes will be awarded: Fulkerson Prizes (for outstanding papers in
discrete mathematics), Orchard-Hays Prize (for excellence in computational
mathematical programming), A.W. Tucker Prize (for an outstanding
paper by a student).
Social program
In addition to the official program, social activities will be organized for the
participants, their family members and friends. This will include a lake
cruise and banquet, visits to museums, a concert, an excursion to a cheese
factory, wine-tasting, sightseeing, hiking, etc....









Preregistration form for ISMP 97


We strongly recommend using the electronic form available via WWW.
If this is not possible, please return a copy of the form below to our mailing address: ISMP 97
c/o Prof. Thomas M. Liebling
EPFL-DMA
CH-1015 Lausanne (Switzerland)
L]Mr. []Mrs. LIMiss


LAST NAME:

INSTITUTION:

STREET:


CITY:


FIRST AND MIDDLE NAME:


COUNTRY:


E-MAIL ADDRESS:


URL ADDRESS (WWW):


PHONE NUMBER:



[I7 intend to give a talk


FAX NUMBER:


0I would like to organize a session or a group of sessions


Remarks and suggestions:


ZIP CODE:


If yes, specify topic of the session (see list of topics):






'AF13N4 DCMER19


R E V I E W S


Dynamic Policies of the Firm

by 0. van Hilten, P. Kort,
P.J.J.M. van Loon
Springer Verlag, Berlin, 1993
ISBN 3-540-56125-0

DURING the last 25 years optimal control
theory has developed as a standard
tool in dynamic economics. In par-
ticular, the dynamics of the firm is
now one of the applications par
excellence of the maximum principle.
The aim of the dynamics of the firm is to study the growth
and contraction ofrepresentative firms in a microeconomic

financing, production policies, impact of taxation, techno-
logical progress and environmental constraints. Early
researchers, like Albach (1976), have stressed that the
development of the firm over time can be divided into
different stages or regimes. In order to understand these re-
gimes in a proper way, optimal control theory delivers a
useful framework.
According to Lesourne and Leban (1982) optimal control
of the dynamics of the firm is an indispensable instrument
L1.1. .. r i, .... t I,,n i.-- .h.. ., the

firm and to provide academic teachers with a tool to out-
line the essentials of the firm. One of the aims of the theory
lies in management training. There is a discussion of how
.1 1 i ,:,, .... ,,- : ,, , ,i1 ,r.- I--, ...C.-, { -,, ,,,,,1 .. I
prices of capital goods and investment grants influence
investment decisions. Today, it is important for the man-


ager to know howoptimal investment decisions have to take
into consideration not only taxation and technological
progress but also business cycles and environmental pol-
lution.
As indicated above, the mathematical tool used to derive
optimal policies ofthe firm is Optimal Control Theory. The
maximum principle formulated by Pontryagin, and proved
by his coworkers Boltyanskii, Gamkrelidze and
Mishchenko in the 50s, yields necessary (and sometimes
also sufficient) optimality conditions 1 i I. ,11 a char-
acterization of the optimal solution trajectories. Two ob-
servations are important. First, the approach is qualitative,
i.e. the model functions and solutions are characterized
qualitatively rather than quantitatively. Second, based on
these optimality conditions, an iterative solution procedure,
the so-called path-connecting procedure, is developed. This
provides the possibility for constructing and interpreting
the optimal solution for the entire planning period in an
analytical way.
This book is divided into five parts. PartAprovides a survey
of dynamic theories of the firm and describes several pre-
decessors of the models presented. In Part B the basic model
is explained and used to discuss optimal investment and
financing behaviour. Part C deals with production and
activity analysis. The representative firm has to choose
between production techniques with different character-
istics. In particular, Chap. 8 discusses the 'hot topic' on
environmental pollution and cleaning activities. In Part D
the firm is faced with an 'outside world' changing over time.

business cycle and a stochastic demand function. The rest
of the book contains six appendices, mainly an economic
interpretation of the maximum principle and technical


~


N 48


IPAGE 13


DECEMBER 1995




N 48


details of the solution procedures of previous chapters. The
book uses the so-.. .: .. .. .. I I .' .withpure
state constraints. The last part of the book in which man-
agement problems in a dynamic environment are analyzed
seems to open interesting new possibilities for further re-
search.
This book is the outgrowth of what today might be called
the 'Netherlands school' of the applications of optimal
control theoryto dynamiceconomics. Pier Verheyen, Paul
van Loon, Geert-Jan van Schijndel, Peter Kort, Raymond
Gradus, Onno van Hilten and others are members of this
school concentrated at I II.,,,.- University. Van Loon
(1983) also wrote the first text bookon the dynamics of the
firm. These scholars continued the work of Bensoussan et
al. (1974), Ludwig (1978), Lesourne and Leban (1982).
The reviewer of this book is reminded of the first encoun-
ter with the 'dynamics of the firm.' It was at a seminar at
the Institute forAdvanced Studies in Vienna given by Horst
Albach in the late '70s. He presented the'path connection
method' as the pedagogical method for management train-
ing. Had it not been for this seminar, the book by
Feichtinger and Hart (1986) (published in German), in
which several models of the dynamics of the firm are de-
scribed might nothave been written. In management train-
ing, control theory models are excellent for showing stu-
dents orjunior managers howto combine policies through
1. 1 i 1 C..... .i-F ., .,-, r ThisbookbyvanHilten,
Kort and van Loon is the main reference in this field.
, I,. I . ..., ... I 1. .... who would like to learn the
core theory of optimal control theory applied to econom-
ics, what single book would you recommend? In my view
the book by van Hilten et al. covers central aspects of
dynamic economics at a level accessible to applied scien-
tists. Theory is presented systematically, and only a mod-
est mathematical background is required in keeping with
the target audience.
References

Albach, H., "Kritische Wachstumsschwellen in der
Unternehmens-entwicldung," Zeitschrift fir
Betriebswfrtschaft46, 683-696, 1976.
Bensoussan, A., E.G. Hurst Jr., B. Nislund, Man-
agementApplications ofModern Control Theory,
North-Holland, Amsterdam, 1974.


Feichtinger, G., R.F. Hard, Optimale Kontrolle
ikonomischer Prozesse: Anwendungen des
Maximumprinzip in den Wirtschaftswissenschaften,
de Gruyter, Berlin, 1986.
Lesourne, J., R. Leban, "Control theory and the
dynamics of the firm: a survey," OR Spektrum 4, 1-
14, 1982.
Ludwig, T., Optimale Expansionspfade der
Untemehmung, Gabler, Wiesbaden, 1978.
Loon, P.J.J.M. van, A Dynamic Theory of the Firm:
Production, Finance and Investment, Lecture Notes
in Economics and Mathematical Systems, Vol. 218,
Springer, Berlin, 1983.
-GUSTAV FEICHTINGER


Discrete and Fractional
Programming Techniques for
Location Models

by Anna Isabel Martins Botto de Barros
Tinbergen Institute Research Series 89,
Amsterdam, 1995
ISBN 90-5170-320-1

SN DISCRETE LOCATION, one is interested in locat-
,,1 i[,, .., . .h .., ,, ., of p o in ts.
In fractional programming, one is interested
in optimizing a program where the objective
function is a ratio of a numerator and denomi-
nator function (such functions may also occur
in the constraints). In this monograph, the author integrates
these two concepts to study a variety of models of interest
in location theory.
Chapter 2, entitled Discrete Location Models, reviews the
I II ..... uncapacitated facility location model and its
extensions to two levels and two echelons. In the two level
case, fixed costs are associated with each level; in the two
echelon case the fixed cost in the second echelon is jointly
determined by the first and the second echelon. It is shown
that the submodularity property does not hold for the two
echelon case. The fixed cost structure in both the above cases
is then aggregated into a new model which generalizes the
above. A variety ofbounds using both linear and Lagrangian


relaxation are established. Together with heuristics they are
used in a branch and bound algorithm, and extensive
computational results are given.
Chapter 3 integrates ideas in fractional programming with
the location models discussed in Chapter 2. It is explained
that the traditional criterion of maximizing profit is some-
times replaced by a profitability index defined by the ratio
of the present value of a project and the investment made.
A brief overview of fractional programming is then made
with emphasis on Dinkelbach's algorithm and its extension
to in tegerl.. .- . '.. 1i.; 1 .;,, I rl , .1. C .. .. I
lates the location models in Chapter 2 with a ratio objec-
tive asdescribedabove. I ,. I ... t ..., 1 .1 II..
some structural results are given.
The author moves her focus from discrete optimization
models in location to generalized fractional programming
for continuous variables in Chapter 4. The chapter begins
by reviewing ideas of Crouzeix et al. that extend
Dinkelbach's approach to generalized fractional programs.
An example involving allocation to a distributed service
network is then given. This motivates the study of gener-
alized fractional programming with a constraint on the
I ..... . '. I II 1 . 1 ..i I, .. ,I . developed.
The author then turns to the dual problem as an alterna-
tive vehicle to solve generalized fractional programs. A new
,,1., ,;,1,,,, ; ; ... ,I..... ;1 1, ,,;..,, ... . n... .. . ,,h ,
once again extending Dinkelbach's parametric ideas. A
further generalization to nonlinear convex constraints is
made. I was particularly pleased to see that algorithms in
this chapter rely heavily on versions of the dual program.
The chapter concludes with extensive computational test-
ing of ratios of positive definite quadratic functions and
affine functions.
In summary, the book should be interesting to those in
fractional programmingwho are interested in seeingwhere
the theory can be applied. It should also be interesting to
those in location who may not be aware of the potential of
theoretical and computational ideas from fractional pro-
gramming in their field. Overall, it is a valuable addition
to the literature of the field.
-CARLTON SCOTT


~--- ~---- ~-"


DECEMBER 1995


PAGE 14





PAGE 15
...... .. ... ...... . . . . .. . . ... .. ...


S


nlumber


N.. 1. .
.\.ll~l.lli II Ir 1,111




...... I aard.,I.i r. rd



1 1I l .I I

H o~s qItech d








la 1. I I I ia ,


U\I1 irror 1-1~ k ni O I



fo I I i..Il Z11i1'I1 II1


DECEMBER 1995

Arbt t- fall I iFOR.iX niIeeinf .
the I anchesitr Prize \'.as\ awarded
ut Richard W. Cortle. Jong-Shi
Pang and Richard E. Stone for
their book, The Linear
Complementarir' Problem. See
OPTIANL N',5 tor a review.
There vwill not be a separate
mailng of the first announce-
ment fo:r the 199- International
Symnposium in Lausanne. It is
fuLind on Pages 11 & 12 of
this issue. ,Members arc urged
to register for subsequent
announcements. !Deadline
for the next OPTINL is
Feb. 15 I'-96.


r - ** ,
S. .



I wish to enroll as a member of the Society.

My subscription is for my personal use and not for the benefit of any library or institution.

D I willpay my membership dues on receipt ofyour invoice.

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