Title: Optima
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00090046/00049
 Material Information
Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: March 1996
 Record Information
Bibliographic ID: UF00090046
Volume ID: VID00049
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.


This item has the following downloads:

optima49 ( PDF )

Full Text






CQlompTl ltarlional Acd,' aces

Usin' the Reformllatiorn-

Lilearizatilon Techmiq ue

(RLT) to Solve Discrete

and Continuous

Nonconvex Problems

Hanif D. Sherali
Department of Industrial & Systems Engineering,
Virginia Polytechnic Institute & State University

Warren P. Adams
Department of Math Sciences
Clemson University

Introduction. Discrete and continuous nonconvex programming problems arise in a
host of practical applications in the context of production planning and control,
location-allocation, distribution, economics and game theory, process design, and
Design situations. Several recent advances have been made in the develop-
ment of branch-and-cut algorithms for discrete optimization problems and in polyhe-
dral outer-approximation methods for continuous nonconvex programming problems.
At the heart of these approaches is a sequence of linear programming relaxations that
drive the solution process, and the success of such algorithms is strongly tied in with
the strength or tightness of these relaxations.
This article describes the application of a Reforiiiilation-Liiinariiatiin-Teichiiqii' (RLT) that
has been developed for generating such tight linear programming relaxations for not
only constructing exact solution algorithms but also to design powerful heuristic
procedures for large classes of discrete combinatorial and continuous nonconvex
programming problems. (ur work initially focused on 0-1 and mixed 0-1 lini)Cr iand
jl/yn(mial programs i : 171, anid later branched into the more general family of
continuous, inoncoilnvx Iofllnial probhiis. (An earlier survey appeared in
I14].) For the family of mixed 0-1 polynomial programs in n 0-I variables, we have ([II,
12]) developed an n-level hierarchy, with the ii-th level providing an explicit algebraic
characterization of the convex hull of feasible solutions. The RLT essentially consists of
two steps a reformulation step in which additional nonlinear valid inequalities are
automatically generated and a linearization step in which each product term is
replaced by a single continuous variable. The level of the hierarchy directly corresponds
to the degree of the polynomial terms produced during the reformulation stage. Hence,
in the reformulation phase, given a value of the level d e 1, . ., ii, RLT constructs
various polynomial factors of degree d comprised of the product of some di binary
variables v or their complements ( x). These factors are then used to multiply each of
the defining constraints in the problem (including the variable bounding restrictions), to
create a (nonlinear) polynomial mixed-integer zero-one programming problem. Suitable
additional constraint factor products can be used to further enhance the procedure. (In
general, for a variable v restricted to lie in the interval i I,'jI, the nonnegative expres-
sions (., -') and (1 X.) are referred to as omnild fhctori:, and for a structural p, l

journals 7
conference notes 8
in memorial 12
book reviews 14


continuedComputational Advances Using RLT

inequality (o. x /3 for example,
thei expression (/ v _i /3 > 0 is
referred to as a coislrain'i l factor.)
After using the relationship rx = .-
for each binary variable x, = .
ii, the linearization phase
substitutes a single variable wii and
(v, respectively, in place of each
nonlinear term of the type i x,,
and l/y ., where i/ represents the
set of continuous variables. I ence,
relaxing integrality, the nonlinear
polynomial problem is linearized
into a higher dimensional
polyhedral set X defined in terms
of the1 original variables (x, Yi) and
the new variables (w, I). Denoting
the projection of X" onto the space
of the original (x, l )-variables as
X,f, it is shown that as d varies
from I to i, we get,

XP iX ,, X ;... ;.. = conv (X)
where X,,, is tile ordinary linear
programming relaxation, and
conv (X) represents the convex
hull of the original feasible
region X.
The hierarchy of higher-dimen-
sional representations produced in
this manner markedly strengthens
tie usual continuous relaxation, as
is evidenced not only by the fact
that the convex hull representation
is obtained at the highest level but
that in computational studies on
many classes of problems, even
the first level linear programming
relaxation helps design algorithms
that significantly dominate
existing procedures in the
literature, producing tight lower
bounds and often times yielding
an optimal solution. Hence, this
general approach holds the
promise for exploiting available
linear programming technology to
effectively solve larger and more
difficult nonconvex problems than
previously possible.
Moreover, the theoretical implica-
tions of this hierarchy are
noteworthy; the resulting
representations subsume and
unify many published lineariza-
tion methods for nonlinear 0-1
programs, and the algebraic
representation available at level ti
promotes new methods for
identivfying and characterizing

facets and valid linear inequalities
in tile original variable space as
well as for providing information
that directly bridges the gap
between discrete and continuous
sets. Indeed, since tile level-I
formulation characterizes the
convex hull, all valid inequalities
in the original variable space must
be obtainable via a suitable
projection; thus such a projection
operation serves as an all-
encompassing tool for generating
valid inequalities. In this spirit, a
framework for characterizing
classes of facets through a
sequential lifting procedure for the
Boole'na quadric polyilopc 1261 has
been devised, with new classes of
facets subsuming the known
clique, cut, and generalized cut
inequality facets emerging. In
addition, new classes of facets
have been characterized tor the
GLIB knapsack !pol/ilope through a
polynomial-time sequential-
simultaneous lifting procedure
Si Known lower hounds on tihe
coefficients of lifted facets derived
from minimal covers associated
with the ordinary knapsack
polytope have been tightened. For
tile set partilioiinlg iolylope ..-I a
number of published valid
inequalities along with constraint
tightening procedures have been
shown to be automatically
captured within the first- and
second-level relaxations thelm-
selves. A variety of partial
applications of the RLT scheme
have also been developed in order
to delete fractional linear pro-
granmming solutions while
tightening the relaxation in tihe
vicinity of such solutions.
The hierarchy of relaxations
emerging from the RLT can be
intuitively viewed as "stepping
stones" between continuous and
discrete sets, leading from the
usual linear programming
relaxation to the convex hull at
level-n. By inductivelv progressing
along these stepping-stone
formulations, we have studied
sorne novel 'prsifsitciici/ issues for
certain constrained and uncon-
strained pseudo-Boolean pro-
granmming problems. Given the

tight linear programming
relaxations afforded by RLT, a
pertinent question that can be
raised is that if we solve a
particular d"' level representation
in the RLT hierarchy, and some of
the ii variables turn out to be
binary valued at optimnality to tihe
underlying linear program, then
can we expect these binary values
to "persist" at optimality to the
original problem? Iln we derive
sufficient conditions in terms of
tlie dual solution that guarantee
such a persistency result. For the
unconstrained pseudo-Boolean
program, we show that for il = I or
for id 2 il 2, persistency always
holds. loiwever, using an example
with d = 2 and it = 5, we show that
without the additional prescribed
sufficient conditions, persistency
will not hold in general. These
results are also extended to
constrained polynomial 0-1
programming g problems. In
particular, the analysis here
reveals a class of 0-1 linear
programs that possess the
foregoing persistency property.
Included within this class as a
special case is the popular vertex
packing problem, shown earlier in
tie literature to possess this
Recently 115], we have extended
our RLT framework to generate a
new hierarchy of relaxations
leading to the convex huIll
representation based on the
development of more generalized
product factors, other than simply
x and (1 x), for I = 1, ..., in the
reformulation phase. In addition,
this hierarchy embeds within its
construction stronger logical
implications than only x2 = x VI =
, ..., ii. As a result, it not only
subsumes the previous develop-
ment but also provides the
opportunity to exploit frequently-
arising special structures such as
generalized/variable upper
bounds, covering, partitioning,
and packing constraints, as well as
Although the Reformulaltion-
Linearization Technique was
originally designed to employ

factors involving iiro-onti variables
in order to generate zero-one
(mixed-integer) polynomial
programming problems that are
subsequently re-linearized, the
approach has also been extended
to solve continuous, bounded
variable polynomial programming
problems. Problems of this type
involve the optimization of a
polynomial objective function
subject to polynomnial constraints
in a set of continuous, bounded
variables, and arise in numerous
applications in engineering
design, production, location, and
distribution problems.
In [291 we prescribe an RLT
process that employs suitable
polynomial factors to generate
additional polynomial constraints
through a nmltiplication process
which, upon linearization through
variable redefinition, produces a
linear programming relaxation.
The resulting relaxation is used in
concert with a suitably designed
partitioning technique to develop
an algorithm that is proven to
converge to a global optimum for
this problem. While RLT essen-
tially operates on polynomial
functions having integral expo-
nents, many engineering design
applications lead to polynomial
programs having general rational
exponents. For such problems, we
have recently developed a global
optimization technique I101 by
introducing a new level of
approximation at the reformula-
tion step and, accordingly,
redesigning the partitioning
scheme in order to induce the
overall sequence of relaxations
generated to become exact in the
limit. Our ongoing investigations
include the extension of the RLT
theory to accommodate discrete-
valued variables in general, as
opposed to the more restrictive 0-1
case. We have already verified that
an analogous hierarchy results,
once again leading to the convex
hull representation at level n, with
a paper forthcoming. (This was
presented at the INI-'ORMS
Meeting, New Orleans, Fall, 1995.)


MARCH 1996



'l'e insights obtained in the
development of the RI T have also
led to a variety ol reIlated linear-
zation results. FoIr example, the

functions known as "paved upper
iplaine'S and "roots" were
explained in terms of a
iLagrangean dual to the level-one
linearization, with the bound
called the "height" being the
optimal objectwive function value to
this formulation 121, An offshoot ofl
this studv allowed for the
extension of the published
persistency results on "roof
duality" to polynoinial pseudo-
iooleatn unctions U11 In tact, the
persistency results on the hierar-
chical levels led to the develop-
ment ol anll entire new lineariza-
tion strategy which, while not
producing convex hull representa-
tions, characterize an entire family
of persistent linear reformulations
of various constrained and
unconstrained 0-1 polynomial
programs l[9, ene'ompassing and
generalizing all known persistent
Over the remainder of this article,
.we now focus on some specialized
RLT designs that have been used
to solve various specific discrete
and continuous noncconlvex
programming problems, and
relate our computational experi-
ence obtained in these instances.
Zero-orne quadratic programs
and the quadratic assignment
problem. The zero-one quadratic
programming problem seeks to
minimize a general quadratic
function in n 0- variables, subject
to linear equality and/or inequ'ial-
itv constraints. In [\i we presented
a new lineari/ation technique' that,
in effect, evolved to become
precisely the level-one relaxation
of the RLT hierarchy discussed
above. 'This relaxation was shown
Ito thorehcaliv dominate other
existing linearizations and was
shown to computationallv
produce far tighter lower bounds.
In these computations, we solved
quadranc set c covering problems

having up to 70 variables and 410
constraints. i For example for this
largest size problem, where the
optimum objective value was
1312, our relaxation produced an
initial lower bound of 1289 at thei
rool node, and enumerated 14i
nodes to solve the problem in 7[ )
cpu seconds on an IBM 308' Series
1)24 group p K computer. When the
same algorithmic strategies were
used on a relaxation that did not
include the special RLT con-
straints, the initial lower bound
obtained was 398, and tihe
algorithm enumerated 2130 nodes,
consuming 197 cpu seconds.
The first and second level RTI
relaxations have also been used to(
develop strong lower bounds for
the quadratic assignment problem.
Because the assignment con-
straints are equality restrictions,
these RLT relaxations are pro-
duced by simply multiplying the
constraints with individual or
with pairs of variables, respec-
tively, as pointed out in Section 7
of I 12t. In 13, we show that the
lower bound produced by the first
level relaxation itself subsumes a
multitude of known lower
bounding techniques in the
literature, including a host of
matrix reduction strategies. By
designing a heuristic dual ascent
procedure for the level-one
relaxation and by incorporating
dual-based cutting planes within
an enumerative algorithm, an
exact solution technique [41 has
been developed and tested that
can competitively solve problems
up to sive 17. In an effort to make'
this algorithm generally appli-
cable, no special exploitation of
flow and/or distance svnmmnetries
\vwa, considered. As far as the
strength of the RIAT relaxation is
concerned, on a set of standard
test problems of sizes 9-20, the
lower bounds produced by the
dual ascent procedure uniformly
dominated 12 other competing
lower bounding schemes except
for one problem of size 20, where
our procedure yielded a lower
bound of 21412, while an eigen-
value-based procedure produced a
lower hound of 2229, the optmrnum

value being 2570 for this problem.
Recently, Rese'nde c' al. (()Opc'iioni
RIi'ii-h, IW995) have been able to
solve the first level RIT relaxation
exactly for problems of size up to
30 using an interior-point method

that employs a preconditioned
conjugate gradient technique to
solve the system of equations for
computing the search directions.
(For the aforementioned problem
of size 20, the exact solution value
of the lower bounding RLT
relaxation turned out to be 2182,
conlmpared to our dual ascent value
of 21423 As a point of interest, we
mention that Ramachandran and
Pekny (INFORMS, FIall 1995) have
been conducting research on
precisely the second and higher-
level RLT7 relaxation ior this
problem, promoting encouraging
preliminary results.
We have also applied RLTI to the
problem ot assigning aircraft to
gates at an airport, with the
objective ofl minimizing passenger
walking distances 11S Il The
problem is modeled as a variant of
the quadratic assignment problem
with partial assignment and set
packing constraints. The quadratic
problem is then equivalently
linearized by applying the first-
level of the RILT. In addition to
simply linearizing the problem,
the application of this technique
generates additional constraints
that provide a tighter linear
program I mming representation.
Since even the first-level relaxation
can get quite large, we investigate
several alternative relaxations that
either delete or aggregate classes
of RLT constraints. All these
relaxations are embedded in a
heuristic that solves a sequence of
such relaxations, automatically
selecting at each stage the tightest
relaxation that can be solved with
an acceptable estimated effort, and
based on the solution obtained, it
fixes a suitable subset of variables
to 0-1 values. This process is
repeated until a feasible solution is
constructed. The procedure was
computationally tested using
realistic data obtained from LISA/li
for p robles having up to 7 gates
and 3(, flights. For all the test

problems ranging Irom -4 gates
and 30 flights to 7 gates and 14
flights, ior which Ihe size ot the
first-level relaxation was manaige-
able (having 14, 494 and 4,084
constraints, respectively, fr t these
two problem sizes), this initial
relaxation itself always produced
an optimal 0-1 solution.
Continuous and discrete
bilinear j ... ,. 'ii ii
problems. lie well-known
nonconvex, NP-hard bilinear
programming problem seeks to
Minimci qe (r, y/) c' + d i/ + .'C;/,
sldhjicl lo (x, I/) r Z n i 2, where
xR", /'R'", / is a polyhedron in
R""", and 2 is a ihyperrectangle,
representing finite lower and
upper bounds oni the variables.
The problem considered is
sa'irabiy constiraino if Z is
separable over x and ,i/, as is often
assumed to be the case, and is
loiiily cistraiiiind, otherwise. (The
latter class of problems are more
difficult to solve in practice.)
Problems of this type find
numerous applications in
economics and game theory,
location theory, dynamic assign-
ment and production problems,
and various risk management
An enhanced first-level RLT
relaxation is designed in 117] lor
such problems by using all
pairwise products of structural
and bounding constraint factors.
For special classes of polytopes,
this is shown to vield an exact
convex hull representation. (Also,
see 1(1I.) More generally, this
yields a linear programming
relaxation that is embedded
within a provably convergent
bra nch-and-bound algorithm.
Computational experiments were
conducted on problems of size up
to 14 variables, including both
separably and jointly constrained
test problems from the literature,
as well as on randomly generated
problems. For all 15 instances of
separably constrained problems
and a great majority oA jointly
constrained problems (i5 oult of 18
instances), the initial linear
programming relaxation itself

MARCH 1 ")0

.,i Ad N" 49

continuedCornputational Advances Using RILT

solved the underlying bilinear
problem. Whenever this was not
the cast, tlMe initial gap between
the lower and upper bounds was
close enough to produce an
optimum after enumerating only a;
few nodes (tewer than I ). This
performance exhibits a significant
improvement over tile previously
best algorithm based on convex
envelopes, whichh consumed tar
more effort and was unable to
solve several of tle test problems
within the set computational
lim its.
In we consider a variation of
the bilinear progranummn problem
il which one of the sets of
variables is restricted to be binary
valued, representing discrete
location or investment decisions,
and where the continuous and the
binary variables are separably
constrained. A partial, modified,
first-level RLT relaxation is
constructed in which houlnd
Actors based on one set of
variables are appropriately used to
multiply constraints involving the
other set of variables, followed bvi
a linearization of the cross-product
terms. The proposed algorithms
additionally employs Benders'
cutis, disjunctive cuts, and
Lagralngan relaxation strategies.
Again, very favorable computa-
tional results have been reported
on an extensive set of test
problems. In particular, problems
having up to 100 continuous
variables and 70 bnaryv variables
were solved to optimality within
about 250 cpu seconds on an IBM
3081 computer. For three different
classes of problems differing in
signs on the objective coefficients
and the density ot the constraints
in the 0-1 variables, the average
initial lower bounds over 6(
llstances of various sizes were,
respectively 99.62'/, 9.24'4, and
83'q l Hence, evident, tlhe success
of thile algorithm is strongly related
to the tightness otf the boundstl
produced by RLT.
In 123, we have applied a similar
RI .' approach to solve general
linear complementarity problems
(.'l') where the underIvin-,

matrix M does not possess any
special property. (Also, see 1221.)
lF'orimating such problems
equivalently as mixed-integer
bilinear programming problems,
the RLTI process described above
for the latter class of problems was
enhanced by Incorporating
conllstraint factor cross-products as
well, and by exploiting the fact
that the optimal objective function
value is zero it and only itf an I.LC'
solution exists. On a total ot 70 Iest
problems using negative definite
and indefinite matrices A4 of size
ilp to 2 x 25, all problem
instances except for one were
solved at the root node itself via
the solution ot a single linear
program. For the one exception,
the LIP solver C(IIHX quit alter
hitting a limit of 10,000 iterations.
I however, when a subgradcient
based .agrangivan dual approach
was applied to this problem, thel
lC' was again solved at the root
node itsel. In general, although
th(e l,agranlgialn dual approach was
unable to attain tile same tight
bounds as C ILEX did due to
convergence difficulties, and as a
result, it sometimes led to an
enumeration of 2 or 3 nodes, it
was still 3-4 times faster in terms
of tile overall effort required an
compared with the C'P.LX based
Continuous and discrete
locatoion-allocationl prob-
le ns. 1The RLT strategy has been
unsecl to derive very effective
algorithms for capacitated,
multifacility, location-allocation
problems that find applications in
service facility or warehouse
location, or manufacturing facility
flow-shop design problems. CGiven
I demand locations customerss or
ntchines) having known respec-
tive demands, the problem is to
simultane'ouslv determine the
ioCi7i'ois oaf soe in supply centers
(service facilities, warehouses,
interacting machines, or tooling
centers) having known respective
capacities, and an aillocailni o(f
products froml each source to each
destination, in order to nininw/e
lotal distribution costs. For the

rcclilim'ar dislancr variant of this
problem that arises in applications
where the flow of goods or
materials occurs along grids of city
streets or factory aisles, the cost is
directly proportional to the
shipment volume and the
rectilinear distance through which
this shipment occurs. This
problem can be equivalently
reformulated as a mixed-ilneger,
zero-one, bilinear programming
problem of the general ftrln
studied in I|8. Wie specialized the
discussed level-one RI T proce-
dire for ltie above problem in I271
and were able to solve these(
difficult nonconvex probtlens
having up to 5 sources and 20
customer locations to opti'mality.
In addition, because of the light
relaxations obtained, this algo-
rithm also provides an efficient
heuristic which iuplo premiatutret
termination is capable i of btailninl
provably good quality solutions
(within 5-10',i of optimality) for
larger sized problems.
Another interesting location-
allocation problem arises in the
case when the per unit transporta-
tion cost penalty is proportional to
the squared Euclidean distance
between the supply and destina-
tion points.l In contrast with the
transformation used for the
rectilinear distance problem that
essentially analyzes that problem
over the location decision space,
we 1301 projected the squared
iouclidean distance problem onto
the space of the allocation-
variables alone, transforming it
into one of minimizing a concave
Lquladratic function over the'
transportation constraints. For this
equivalent representation, _we
devised a suitable application of
the RLT concept by generating a
selective set of first level houndt
factor based RLI constraints. Our
computational tests have revealed
that the bounds obtained from this
relaxation are substantiallY
superior to four different lower
bounds obtained using standard
techniques,. Computational
experience reveals that the initial
linear program itself produces
solutions within 2-4''i of

optinmahlt and that this procedure
significantly enhances the size (o
problems solvable by a branch-
and-bound algorithm. We have
solved problems having (i0,i)-
(0,120) (20,60) within about 150
cpul sees onil anll IBM 3090 corn-
puter, while the methods, eumploy-
ing four standard lower bounding
lechniclues (previously developed
bv others) were able to handle
problems oi size up to only I -
and H = 6 withiAn 370 cpu sees on
the same compu ter.
Predating this work, we had also
studiedd a discrete variant of the
location-allocation problem in
which the iii capacitaed service
facilities are to be assigned in a
one-to-onle fashion to some i
discrete sites itn order to serve the
i1 custontrs, Vwihere tihe cost per
unit flow is determmed by stme
general lacilitv-customer separa-
tion based penalty Iunction 1131
This problem also turns out to
have the structure ol a separably
constrained mixed-integer bilinear
protgramlming problem, and a
partial first level RI T relaxation
that includes onlv a subset of the
constraints developed iln 81, sotme
1n aln agg reiat'ed form, was used
to generate lower bounds. A set )o
I1 problems with (hi,St) ranging up
to (7,50)-(1 1,11) were solved using
a Benders' partitioning approach.
For these problem instances, even
the partial, aggregated first level
RLT relaxation produced lower
bounds within 90-9'/ of
Indefinite quntdratic pro-
granms and poyhnoimial
progranmming problems. In
I31 ], we have developed a global
optimization procedure for
linearly constrained indefinite/
concave-minimnization quadratic
programmlling problems. These are
hard noncoiivex programming
problems that can have ma1ny local
optima that differ significantly
from the global optimal solution,
We have designed and tested RILT.
based relaxations for such
problems which depart somewhat
from previous approaches in that
the relaxations are nlot purely

MARCH 199')

--------rm 1Irr1 11

linear, but they retain a critical,
manageable degree of nonlinearity
in terms of separable quadratic
functions. Specifically, a selected
set of first level bound factor
based as well as constraint factor
based RLT constraints are
generated and are augmented by
convex, variable upper bounding
restrictions of the type xs < for
each variable x, in the problem,
where is the linearized variable
representing the nonlinear term x1.
Furthermore, various range-
reduction strategies were devised
based on the eigenstructure of the
problem, logical tests on the
constraints, and the Lagrangian
dual based lower bound produced
via the RLT relaxation. These
strategies were all embedded in a
branch-and-bound framework to
provide global optimal solutions
to the underlying nonconvex
problems. The tightness of the
RLT relaxations was evident from
the computational results
presented on solving several test
problems from the literature of
size up to 20 variables. For many
instances, the problem was solved
at the initial node itself, while for
others, only a few nodes were
enumerated in solving the
problem. Among the former
instances of problems was one
particularly notorious jointly
constrained bilinear program that
had required 11 nodes to be
enumerated by our previous RLT
based algorithm for this class of
problems and over 105 nodes for a
competing convex envelope based
approach. Also, one published test
problem in 20 variables had not
previously been solved to opti-
nality. In addition, we attempted
to solve a set of larger (up to 50
variables) randomly generated
problems obtained using an
available test problem generator.
In all the seven instances solved,
the lower bound obtained at node
zero was within 95.2-)9.9'; of
optimality. Fence, this technique
affords a tool to derive near
optimal solutions to hard
nonconviex problems via a single
(essentially linear) relaxation.

We have also devised similar RLT
based strategies to solve polyno-
mrial programming problems. A
specialization to the water
distribution network design
problem is described in [281 where
global optimal solutions to a
standard test problem from the
literature, and some of its variants,
are presented for the first time
ever. (A recent paper by C. Eiger,
U. Shamir and A. Ben-Tal in Water
Resources Rsearchi, 30(8), 2637-
2646, provides the only other
global optimization approach for
this problem, and also solves
certain other instances to
optimality.) For general polyno-
mial programming problems, we
present in 1321 some theoretical
and computational dominance
results of applying the RLT lower
bounding scheme directly as in
[29) to the original polynomial
program, versus applying it to an
equivalent quadratic polynomial
program that is obtained through
a standard successive transforma-
tion process. This dominance
holds even if all alternative ways
of making such a transformation
are siiiitaniously included within
the quadratic reformulation.
In 1331, we present various new
classes of RLT constraints that can
be used to tighten relaxations. For
univariate polynomial programs,
these constraints use certain
special squared grid factor
products based on a discretization
of the bounding interval in one
case and certain squared Lagrange
interpolation constraints based
also on a discretization in the
second case. In each case, the
stated constraints define valid
inequalities derived by suitably
composing products of factors,
where each factor is based on a
grid point rather than simply on a
bounding interval end-point. For
example, on a sample of problems
having degree 3, 4, and 6 with
respective optimal values -4.5, 0,
and 7, respectively, our RLT
procedure produced the optimum
via the initial lower bounding
linear program itself, while a
standard exponential transfortma-

tion based technique produced
initial lower bounds of -1311, -
22354, and -5519681, respectively.
Similarly, we solved some
standard constrained multivariate
engineering design test problems
from the literature having ni
variables, and polynomial
objective and constraint functions
of degree S. Again, the tightness of
the lower bounds produced by the
RLT linear programming relax-
ation at the root node was
instrumental in algorithmic
efficiency. For example, a set of
problems having (n, 8) = (2, 4), (4,
4), (10, 3), and (13, 2) with
respective optimal values of -
16.7389, 17.01417, -1768.807, and
97.588, respectively, the initial
lower bounds produced by the
RLT relaxation were -16.7389,
16.75834, -1795.572, and 97.588,
respectively. Hence, only a few
nodes (often just 1-3) were
required to solve several of these
problems to 1% of optimality.
Conclusions and extensions.
In this article, we have described
our experience in designing
specialized RLT based relaxations
for solving various specific classes
of discrete and continuous
nonconvex problems. We are
currently investigating RLT
designs for many other applica-
tions, notably, some telecommuni-
cation design problems as well as
the development of general
purpose algorithmic strategies for
solving linear mixed-integer 0-1
programming problems and
continuous polynomial program-
ming problems. Specific special
structures inherent within such
problems can be exploited by the
RLT process as discussed in [115].
In conclusion, there is one
particular helpful comment that
potential users of RLT need to
bear in mind. As evident from our
discussion, in using the RLT
approach, one has to contend with
the (repeated) solution of large-
scale linear programming
problems that are generated to
provide tight polyhedral outer-
approximations for the convex

hull of feasible solutions. By the
nature of the RLT process, these
linear programs possess a special
structure induced by the repli-
cated products of the original
problem constraints (or its subset)
with certain designated variables.
At the same time, this process
injects a high level of degeneracy
in the problem since blocks of
constraints automatically become
active whenever the factor
expression that generated them
turns out to be zero at any feasible
solution. As a result, simplex-
based procedures and even
interior-point methods experience
difficulty in coping with such
reformulated linear programs (see
18] for some related computational
experience). On the other hand, a
Lagrangian duality based scheme
can not only exploit the inherent
special structures but can quickly
provide near optimal primal and
dual solutions that serve the
purpose of obtaining tight lower
and upper bounds. References [9),
20, 21, 341 provide a discussion on
related literature, as well as new
subgradient deflection and
stepsize techniques and proce-
dures, for generating both primal
and dual solutions via such a
Lagrangian dual based approach.
Acknowledgement. This
material is based upon research
supported by the National Science
Foundation under Grant Number

References Related to
Authors' RLT Research
Ill W. P. Adams, A. Billionnet, and
A. Sutter, "Unconstrained 0-1
Optimization and Lagrangean
Relaxation," Discrete Applied
Mathematics, Vol. 29, Nos. 2-3, pp.
131-142, 1990.
121 W. P. Adams and P. M. Dearing,
"On the Equivalence Between Roof
Duality and Lagranian Duality for
Unconstrained 0-1 Quadratic
Programming Problems," Discrete
Applied Matheimatics, Vol. 48, No.
1, pp. 1-20, 1994.


N', 49

MARCH 1996

,v;i 6 N" 49

continuedComputational Advances Using RLT

13] W. P. Adams and T. A. Johnson,
"Improved Linear Programming-
Based Lower Bounds for the
Quadratic Assignment Problem,"
DIMACS Series in Discrete
Mathenmtics and Ttheoretical
Computer Science, "Quadratic
Assignment and Related Prob-
lems," eds. P. M. Pardalos and LH.
Wolkowicz, 16, pp. 43-75, 1994.
14] W. P. Adams and T. A. Johnson,
"An Exact Solution Strategy for the
Quadratic Assignment Problem
Using RLT-Based Bounds,"
Working paper, Department of
Mathematical Sciences, Clemson
University, Clemson, SC, 1996.
151 W. P. Adams, J. B. Lassiter and
H. D. Sherali, "Persistency in 0-1
Optimization," under revision for
Mathematics of Operations
Research (Manuscript, 1993).
16] W. P. Adams and H. D. Sherali,
"A Tight Linearization and an
Si ..i;li for Zero-One Quadratic
Programming Problems," Manage-
ment Science, 32(10), pp. 1274-1290,
171 W. P. Adams and Li. D. Sherali,
"Linearization Strategies for a Class
of Zero-One Mixed Integer
Programming Problems," Opera-
tions; Research, 38(2), pp. 217-226.
18] W. P. Adams and H. D. Sherali,
' I ..1 in r- Bilinear Program-
ming Problems," Matlhematical
Progrant inig, 59(3), pp. 279-306,
191 J. B. Lassiter and W. P. Adams,
"Strategies for Converting Various
0-1 Polynomial Programs into
Persistent Mixed 0-1 Linear
Problems," '.... I.... paper,
Department of Mathematical
Sciences, Clemson University,
Clemson, SC, 1994.
1101 H. D. Sherali, "Global
Optimization of Nonconvex
Polynomial Programming Problems
Having Rational Exponents,"
Working paper, Department of
Industrial and Systems Engineer-
ing, Virginia Polytechnic Institute
and State University, I I,. I ....
. .... 24061, 1996.

1111 H. D. Sherali and W. P. Adams,
"A fHierarchy of Relaxations
Between the Continuous and
Convex Hull Representations for
Zero-One Programming Problems,"
SIAM Journal on Discrete Math-
ematics, 3(3), pp. 411-430, 1990.
112] H. D. Sherali and W. P. Adams,
"A Hierarchy of Relaxations and
Convex Hull Characterizations for
Mixed-Integer Zero-One Program-
ming Problems," Discrete Applied
Mathematics, 52, pp. 83-106, 1994.
1131 H. D. Sherali and W. P. Adams,
"A Decomposition ,.i ... ;iii o for a
Discrete Location-Allocation
Problem," Operations Research, pp.
878-900, 1984.
114] H. D. Sherali and W. P. Adams,
"A Reformulation-Linearization
Technique (RLT) for Solving
Discrete and Continuous
Nonconvex Programming Prob-
lems," Vol. XII-A, pp. 61-78,
Mathematics Toiday, special issue
on Recent Advanceis in Mattematni-
cal P1ro'granmoiinrg, ed. O. Gupta,
1151 H. D. Sherali, W. P. Adams,
and P. Driscoll, "Exploiting Special
Structures in Constructing a
Hierarchy of Relaxations for 0-1
Mixed Integer Problems," Working
paper, Department of Industrial
and Systems Engineering, Virginia
Polytechnic Institute and State
University, I I I.. .... VA 24061-
0-118, 1994.
1161 H. D. Sherali and A.
Alameddine, "An Explicit Charac-
terization of the Convex Envelope
of a Bivariate Bilinear Function
Over Special Polytopes," Annlals of
Operations Research, Compmtal-
tional Methods iln Global Optimiza-
tion, eds. J. B. Rosen and IP.
Pardalos, 25(1-4), pp. 197-214, 1990.
1171 H. D. Sherali and A.
Alameddine, "A New Reformula-
tion-Linearization Technique for
..i ,.. Bilinear Programming
Problems," Journal of Global
Optimization, 2, pp. 379-410, 1992.
1181 Ht. D. Sherali and E. L. Brown,
"A Quadratic Partial Assignment
and Packing Model and Algorithm
for the Airline (ate Assignment
Problem," I)IMACS Series inl
Discre'te Mathemalics and Theoreii-

cal Computer Science, "Quadratic
Assignment and Related Prob-
lems," eds. P. M. Pardalos and H.
Wolkowicz, 16, pp. 343-364, 1994.
1191 H. D. Sherali, G. Choi, and Z.
Ansari, "Memoryless and Limited
Memory Space Dilation and
Reduction Algorithm," under
revision for the Journal of Optimi-
zation Theory and Applications.
1201 H. D. Sherali, G. Choi, and C.
H. Tuncbilek, "A Variable Target
Value Method," under revision for
Mathemalical Programming.
1211 H. D. Sherali and G. Choi,
"Recovery of Primal Solutions
When Using '- I, 'l. i ... Optimi-
zation Methods to Solve
Lagrangian Duals of Linear
Programs, Operations Researchl
Letters, to appear.
1221 H. D. Sherali, R. S.
Krishnamurthy and F. A. Al-
Khayyal, "An Enhanced Intersec-
tion Cutting Plane Approach for
Linear Complementarity Prob-
lems," journal of Optimizationt
Theory and Applications, to appear.
1231 H. D. Sherali, R.
Krislmamurthy and F. A. Al-
Khayyal, "A Reformulation-
Linearization Approach for the
General Linear Complementarity
Problem," Presented at the joint
National ORSA/TIMS Meeting,
Phoenix, Arizona, October 31-
November 3, 1993.
1241 H. D. Sherali and Y. Lee,
"Sequential and Simultaneous
Lifting of Minimal Cover
Inequalities for GUB Constrained
Knapsack Polytopes," SIAM
Journal on Discrete Matitemalics,
8(1), pp. 133-153, 1995.
1251 1-. D. Sherali and Y. Lee,
I l,. i Representations for Set
Partitioning Problems Via a
Reformu lation-Linearization
Approach," Discrete Applied
Mathematics, to appear.
1261 H. D. Sherali, Y. Lee and W. P.
Adams, "A Simultaneous Lifting
.i. for Identifying New
Classes of Facets for the Boolean
Quadric Polytope," Operations
Research Letters, 17(1), pp. 19-26,

127] H. D. Sherali, S. Ramachandran
and S. Kim, "A Localization and
Reformulation Discrete Program-
ming Approach for the Rectilinear
Distance Location-Allocation
Problem, "Discrete Applied
Matlhematics, 49(1-3), pp. 357-378,
128] H. D. Sherali and E. P. Smith, "
A Global Optimization Approach
to a Water Distribution Network
Problem," under revision for the
Journal of Global Optimization.
129 11. D. Sherali and C. H.
Tuncbilek, "A Global Optimization
Algorithm for Polynomial
Programming Problems Using a
Technique," Journal of Global
Optimization, 2, pp. 101-112, 1992.
1301 H. D. Sherali and C. I.
Tuncbilek, "A Squared-Euclidean
Distance Location-Allocation
Problem," Naval Research
Logistics, 39, 447-469, 1992.
131 HI. D. Sherali and C. iH.
Tuncbilek, "A Reformulation-
Convexification Approach for
Solving Nonconvex Quadratic
Programming Problems," Journal of
Global Optimization,, pp. 1-31,
1321 H. D. Sherali and C. H.
Tuncbilek, "Comparison of Two
Technique Based Linear Program-
ming Relaxations for Polynomial
Programming Problems," Working
paper, Department of Industrial
and Systems Engineering, Virginia
Polytechnic Institute and State
University, I I .. I -l... VA,1995.
1331 H. D. Sherali and C. H.
Tuncbilek, "New Reformulation-
Linearization Technique Based
Relaxations for Univariate and
Multivariate Polynomial Program-
ming Problems," Working paper,
Department of Industrial and
Systems Engineering, X; 1; I
Polytechnic Institute and State
University, Blacksburg, VA, 1996.
1341 H. D. Sherali and O. Ulular, "A
Primal-Dual Conjugate
,i ,.i;. Algorithm for
Specially Structured Linear and
Convex Programming Problems,"
Applied Malthemalics and Optimi-
zation, 20(S-15), pp. 193-221, 1989.

MARCH 1'()

P T6 I M~ E

Volume 71, No. 2

E.H. Aghezzaf, T.L. Magnanti
and L.A. Wolsey, "Optimizing
constrained subtrees of trees."
C. Blair, "A closed-formi
representation of mixed-integer
program' m value functions."
S.E. Karisch and F. Rendl,
"Lower bounds for the quadratic
assignment problem via triangle
A.V. Goldberg and R. Kennedy,
"An efficient cost scaling
algoritihm for the assignment
J.V. Burke and M.C. Ferris, "A
Gauss-Newton method for
convex composite optimization."
R. Sridhar, "Superfluous
matrices in linear
U. Brninnlund, "A generalized
subgradient method with
relaxation step."
E.D. Andersen and K.D.
Andersen, !'* ... ', in linear
, '', 'i'l"" ""

Volume 71, No. 3

M. Conforti and G. Cornuejols,
"Balanced 0,+ 1-matrices,
bicoloring and total dual
S. Filipowski, "On the complex-
ity of solving feasible systems of
linear inequalities specified with
approximate data."
B. De Schutter and B. De Moor,
"The extended linear
complementarity problem."
T.L. Magnanti and G. Perakis,
"A .iii.;,. "geometric solution
framework and complexity
analysis for variational in-
F.B. Shepherd, "Applying
Leihian's theorems to packing
G. Pritchard, G. Giirkan and
A.Y. Ozge, "A note on locally
Lipschitzian functions (Slorl

Nominations fr the A. W Tucker Prize Are Invited
TheMathemnaticallroammingSocietyinvites nominations fr theA.W.
Tucker Prize for an outstanding paper authored by a student. The award will be
presented at the International Symposium on Mathematical in
L.ausainne (24-29 August 1997). All stidients, graduate and undergraduate, are
eligible. Nominations of students who have not yet received the first university
i., II I ... In advance of the Symp osium an award committee
will screen the nominations and select at most three finalists. '- .1I 'I be
invited, but not required, to give oral presentations at a special session of the
Symposium. The award committee will select the winner and present the award
prior to the conclusion oft the Symposium. The members of the committee for
the 1997 A.W. Tucker Prize are: Kurt Anstreicher, Rolf. i. Mohring, Jorge
Nocedal, J.-P. Vial (Chairman) and David Williamson.
Eligibility'Thc paper may concern any aspect of mathematical programming;
it may be original research, an exposition or survey, a report on computer routines
and computing experiments, or a presentation of a new and interesting applica-
rion. The paper must be solely authored and completed aftir anuliary 1 99'i. The
paper and the work on w which it is based should have been undertaken and com-
pleted in conjunction with a degree program.
Nominations must be made in writing to the chairman o the award committee
by a faculty member at the institution where th e tnline was studying tior a degree
when the paper was completed. Letters of nomination must be accompanied by
five copies each of: the student's paper; a separate summI1ary of thie papers con-
tributions, written by the nominee, and no more than two pages in length; and
a brief biographical sketch of the nominee.
Deadline Nominations nmust be sent to
the chairman, as i. II. and postmarked
no later than December 31, 1996: Jean-Philippe Vial
H E .1.. : i .... ..I Studies
University of Geneva
102, Bd Carl Vogt
CH-1211 Geneva 4

Laboratoire Approximation & Optimisation

University Paul Sabatiere
Toulouse, France

i\l Il I I I "i -. d e
l'Optimisation et de la
Decision) is a permanent
group inside SMAI (Societe de
Mathematiques Appliquees et
Industrielles, France) gathering
people interested in: optimiza-
tion (mathematical program-
ming, variational problems,
operations research) and math-
ematics of the decision sciences
(mathematical economics,
mathematics in social sciences).

Its current board committee is
composed of:
J.-B. Hiriart-Urruty (President)
B. Monjardet (Vice-President)
M. Thera (Secretary)
C, Lemar&chal (Treasurer).
The report on activities of 1994 can
be found in MATIAP' I 43 (the news-
letter of SMA1) of July 1995.
For more information please contact:
Address: I 18, route de Narbonne,
31062 TOULOUSE Ccdex France.
Secretariat: Btiiment 1IR2, Porte 25,
rez de chaussee.
Telephone: 61 55 67 78
Telccopic: 61 55 61 83
E-mail: lao@cict-fr
I h I IRIAi i UR I( ; li

N" 49

MARCH 1996

N~' 49

S0 P T I


S5th SIAM Conference on
Optimization, British Columbia
May 20-22, 1996.
S18th Symposium on Math-
ematical Programming with
Data Perturbations
George Washington University
23-24 May 1996
SIPCO V, Vancouver, British
Columbia, Canada
June 3-5, 1996
SFifth International Sympo-
sium on Generalized Convexity
Luminy-Marseille, France
June 17-21, 1996
F 7th Stockholm
Optimization Days
Stockholm, Sweden
June 24-25, 1996
I IFORS 96 14th Triennial
Conference, Vancouver
British Columbia, Canada
July 8-12, 1996
Santa Barbara, California
Aug. 19-23, 1996
SInternational Conference on
Nonlinear Programming
Beijing, China
Sept. 2-5, 1996
SSymposium on Operations
Research (SOR96)
Technical University
Braunschweig, Germany
Sept. 4-6, 1996
SSecond International Sympo-
sium on Operations Research
and its Applications
(ISORA '96)
Guilin, China
Dec. 11-13, 1996
SXVI International
Symposium on Mathematical
Programming, Lausanne
Switzerland, Aug. 1997

con0f eHrencffB

7th Stockholm
Optimization Days
Stockholm, Sweden,
June 24-25, 1996
We welcome theoretical, comIputa-
rional and applied papers for the 7rt
Stockholm Optimization Days, a
two-day conference on optimization,
to be held at K'T1H (Royal Institute
of Technology) in Stoctkholm, Swe-
den, June 24-25, 1996.
'Ihere will be sessions on various as-
pects of optimization, including
ionsmooth optimization, linear and
nonlinear programming, as well as
applications of optimization in areas
StICh as sttrctl ral optim ization alnd
We anticipate soie 30 talks in total,
our of which approximately 15 are
invited presentations.
Abstracts maximumm 200 words)
should be sent by May I (prtcerably
by e-mail) to:
or by mail to:
Optimization Days
Division of O)pimiization a(nd
Systems 'Theory
S-100 44 Stockhoil
Fax: +46 8 22 53 20.
Further information can be obtained
from tie same addresses.
T ihe conference is financially sup-
ported by the G(oran Gustaisson
foundation andl the Swedish Na-
tional Board fior Intdustrial atnd
''eclhical Developmenr (NUT'EK).
lThe organizing committee consists
ofl Ulf Blranilund, Anders Forsgren
and Krisier Svanberg headdi, fom
the Division of )ptimizatilo and
Systems ITheory, Department of
Mathematics, Royal Institute of
TIechnology (KTI I).

PA(&i 8

MARCH 1996

N" 49

SOR96Announcement &d Call fI r Papers
Symposium on Operations Research (SOR96)
Annual Conference of the DGOR and GMOOR with the
participation of WG 7.4 of the IFIP
Technical University
Braunschweig, Germany
September 4-6, 1996

I. Linear -' ...t ', ,
2. N onlinear I .. ........
3. (Comlbinatorial and I)iscrete Optimlization
4. (;raph Algorithms and Complexity
5. Stochastic Models and Optimization
6. Scheduling
7. Production
8. Transportation
9. Macroeconomics, Economic Theory, (names
10. Statistics and Ecionometrics
I 1. Marketing and Data Analysis
12. Information and Decision Support Systeims
13. I ... Fiinance, Ilnsurance
14. Environment, Energy, Health
1 5. Neural Networks and Fuzzy Systems
16. Control Theory
17. Simulation
18. Practical OR (Applicatiotn Reports)
Conference languages: English and (;crman

Apr 1, 1996
Apr 15, 1996

Bixby, Houstoni
Schohtcs, Karlsruhel
BurLkad, (raz
Mohring, Berlin
Mosler, lK6ln
Drexl, Kiel
Tempelmeicr, Koln
Domschke, Darmstadt
iichhorn, Karlsruhli
K reif I i
(iaul, Karlsruhe
Derigs, Koln
Minnemann, Diisseldorf
Stepan, Wien
We(rners, Bochum
Hartl, Wieni
Chamoni, Duisblurg
SchusIctr, Jesteburg

Sending a paper copy oft the abstract
Sending the abstract by c-mail (I i I)

May 15, 1996 Regular registration deadline

JIul 15, 1996 Sendout of the preliminary program
Aug l 1996 I .pI. .. for, 1 ... I1 .... refund
Aug 1, 1996 Receipt of a paper for the proceedings volume

Distribution and gathering of information for SOR96 will to a large extent be
based on e-mail and electronic networks. So, whenever possible, use e-mail and
the Well for communication. IThe e-mail address for all these contacts is:

Certain keywords on the subject linewill
trigger automatic responses, such as
Subject Reply1
info returns the detailed
second IannounIlcemelnt
help returns a description of
the e-mail interface
help returns a registration
registration formi
help abstract returns an abstract form
help hotel returns a hotel form
help returns information
manuscript concerning manuscripts
for the proceedings
To communicate via the Web, start at
and follow the respective links
Organizing committee:
Prof.Dr. R. 1
Prof'.Dr. S. Vofis
Prof.Dr (. G. Wascher
ProEDr.. U. Zinmicrlann
Program committee:
Prof. Dr. U. I)erigs
Prof. Dr. W. Gaul
Prof. Dr. R.H. H IF;,
K.-P. Schuster
Mailing address:
Prof. Dr. U. Zimmetiirmann
Abt. Mathem. Opitiiierung
TU Braunschweig
1)-3810(6 Braunschweig, (;ermanyl
+49(0) 531-391-7550
Fax: +49(0)531-391-7559

bsiekA I Bmo.math.na bs.de

MARCH 1996

Annals of Operations

Special Issue on Parallel
Parallel computing emerges as one of
the most powerful and versatile tools
available to address complex and large
scale problems. Tlihe applications are
varied, IInImerous and meaniinigfil;
one can mention the fields of finance,
real time reaction and decision mak-
ing, i.. I ..I transportation sys-
tens, artificial li, i biology
and chemistry, etc. Yet, tie potential
gains of parallel computation do not
materialize easily. In fact, parallel
comlputation I. researchers to
rethink their models and algorithms,
besides imposing a few specific issues
of its own (e.g., efficient data struc-
tures, performance measures, etc.).
iThere is now a significant body of
scientists in operations research and
mathematical programming actively
involved in addressing these issues
and developing sound and efficient
Parallel Optimizatioon models and al-
gorithms in a wide variety of applica-
tion contexts. We intend this special
issue of the Annahs of Operations Re-
search to capture the essence of
today's state-of-the-art research in
this dynamic and exciting field.
We seek original, high quality contri-
butions that may belong, but are 1not
restricted, to one of the I. .. cat-
* Methodological work on models
and algorithms in continuous lincar
and nonlinear optimization, integer
and mixed integer programming,
combinatorial optimization, network
flows, metabeuristics, global optimi-
zation, stochastic optimization, etc.
PA(;hI 1N "


PAGt;r 10

Issues related to the development,
implementation and evaluation of
parallel optimization methods: load
balancing, data structures, pertor-
mance measures, software libraries,
Applications: transportation, tele-
communication, location, manufac-
turling TSP/VR fi, nance, biology
and chemistry, etc.
To submit a paper, one may send the
manuscript to one of the editors ci-
ther ill paper form five (5) copies
are then required or electronically as
a LaTex file. A specific La'Tex Style
File for the Annals is available either
from the editors or directly from tiec
home page of Baltzer Science Pub-
lishers: http://www.nl.net/~baltzer/
Electronic submissions iare encour-
aged. As an additional incentive, the
publisher offers 50 instead of the nor-
mral 25 free reprints when the authors
use the Baltzer LaTex Style File.
IThe manuscript must be original,
previously unpublished and not cur-
rently under review for other ouri -
nals. To be considered for the special
issue, the manuscript must be re--
ceived by April 15, 1996. All manu-
scripts will be strictly refcrced accord-
ing to the highest standards as out-
liined in the guidelines of the Annals
of Operations Research. For further
information, please contact either of
the guest editors below:
Teodor Gabriel Crainic
Centre de recherche sur les transports
University de Montrdal
C.P. 6128, succ. Centre- II
Montrdal (QC) Canada I3C 3J7
(514) 343 7143
fax: (514) 343 7121
Catherine Routcairol
Laboraroire PRISM
University de Versailles
45, Avenue des lEras-Unis
78035 Versailles Cedex, FIance
(+33 1) 39 25 40 88
fax: (33 1) 39 25 40 57

Second International Sympo-
sium on Operations Research
and its Applications
(ISORA '96)
Guilin, China
December 11-13, 1996
The International Symposium on
Operations Research and its Appli-
cations is a forum for scientists, en-
gineers, and practitioners through-
out the world to exchange ideas atnd
research results related to operations
research and its applications. lihe
first symposium (ISORA '95) was
ield in I China, in August,
1995. The second symposiuml
(1SORA '96) will be lheid December
11-13, 1996, in Guilin, China.
Topics of interest include but are
not limited to: Linear and nonlinear
programming, Combinatoial and
global optimization, Multiobjective
optimization, Stochastic prograim-
ming, Scheduiling acnd network flow,
Queuing systems, Qualiry rechnol-
ogy and reliability. Simulation, Op-
timizations in VLSI, Neural net-
work, Financial modeling and analy-
sis, Manpower planning, Produc-
tion/Inventory control, Flexible
manufacturing systems, Decision
analysis, Decision support systems,
Micro-computer software o OR
methods. Paper s on ireal-world appli
cations will be especially appreci-
I lie symposium is sponsored bhy tie
Asian-Pacific Operations Research
Center within the Association of
Asian-Pacific Operational Research
Societies (APORS) and Ch(inese
Academy of Sciences (CAS).
The symposium chair is Professor
Xiang-Stu Zhang (CAS).
The ISORA '96 is supported by the
Institute of Applied Mathematics,
Chinese Academy of Sciences; the
Operations Research Society of
China; the National Natural Science
Foundation of China; and the Stare
Science and Technology Commiis-
sion of China.

I h. I ,, II ,,,

the country's most beaurtifti scenery.
Authors are requested to submit
five copies of an extended abstract
as follows:
Organizing Commiittee Chair:
Prof. Kan Cheng, Institute of
Applied Mathematics, Chinese
Academy of Sciences, I
100080, P.R. China.
tax: 86-10-254-1689
Program Commnittee Chair:
Prof. D)ing-Zhu I)u, Compulter
Science Department, University of
Minnesota, Minneapolis, MN
55455, U.S.A.
fax: 612-625-0572
Submissions should be written in
English, at most ten pages, and in-
clude thle e-mail address of one au-
thor who is responsible for all corre-
spondence. One author of each ac-
cepted paper should attend the con-
ference and present the paper. Pro-
ceedings of ISORA '96 will be pub-
lished by Beijing World Publishing
Corporation, and selected papers
will be put in a special issue of lthe
Journal of Global Optimization,
The conference welcomes any spe-
cial session on the above topics. Pro-
posals for special sessions should be
sent to one of the addresses above
before July 1, 1996.
I)eadline for submission of papers:
June 1, 1996
Notification of acceptance:
August 1, 1996
Camera-ready manttscript idule:
September 1, 1996
For information about program,
registration and local arrangements,
please contact:
Dr. X.-D. Hu
Fax: 86-10-254-1689
or Prof. I)ing-Zhii Du
fax: 612-625-0572



Il ,1F


i i "

, '-

Szeged, Hungary

December 1995

~ 1 I

MARCH 1996

S\(;i, 1

No' 49

MARCH 1 ''

I ___1 P T I M V S

The Third Workshop
on Global Optimization was
held in December 1995 in Szeged,
IHungary, and was organized by the
Austrian and the Hungarian Opera-
tions Research Societies. More than
60 participants followed a tight
schedule of 45 .' The papers cov-
ered many aspects of the field, such
as new heuristics, utilization of
structural information, method-
ological questions, complexity, effi-
ciency and reliability of global opti-
mization .1 .e'... The reported
applications dealt with such diverse
subjects as protein folding, financial
problems, tracking elementary par-
ticles, chemical process network syn-
thesis, water quality management,
optimal rejuvenation policy, and re-
construction problems in picture
Participants arrived from 19 coun-
tries, including New Zealand, Jor-
dan, Australia, Mongolia, the
United States, (ermany and Russia.
Dlue to te assistance of the sponsors
(Hungarian Research Fund ()TKA,
Veszpreni University, Szeged (:ity
Mayor's Office, Pick Szegcd Rt. and
MO1, Oilindustrial Trust), many
important representatives in the field
were abch to participate.
The papers arising from the talks
presented at the workshop will be
refereied and then published by
Kluwer Academic Publishers in two
special issues: Jourla/ of 'Globa/l Op/i-
iiization, and in the book Develop-
iiieis in Global ()ptimiizaion in the
series Noinronvex Optii/ization ani
ics Application,.
Although the five days were quite
fill, we did find time to see a
dance show and to have a short
sightseeing tout TIhere was also a re-
ce option given by the Rector of trihe
JozsicfAttila UniversityS and another
by tie Mavor of Szeged. Interest-
ingly, the Mayor ofSzeged, Dr.
Istvan Szalay, is a mathematician
working on approximation theory.
The volume of extended abstracts,
photos and Ill;V other d(iocumentsls
of the workshop are available on the
Internet at the I I. .. addresses:
URL: http://www.inf.u-szeged.hu/
FTP: ftp.jate.u-szeged.hu
in the directory /pub/math/
SIIlBo ( St NDI s

Report on the Conference

Gainesville, Florida

inference on Network Optimi-
ition was held at the (Center for Applied
Spitimization at the University ofiFlorida, Feb.
S-14, 1996. This conference was sponsored
the National Science Foundation and en-
dorsed by SIAM, the Mathematical . ".... Society
and the Institute for Operations Research and Management
Science. Organizers were Panos Pardalos, Don Hearn and
Bill Hager.
IThle conference opened with a lec cure by Thomas Magnanti
(MIT) on "Designing Survivable Networks." Often a net-
work needs to be able to withstand disruptions (link or node
failures) and vet still provide service to its customers. T his
can be achieved by building redundancy or spare capacity
into the network or using different types oflinks (or nodes)
with more reliable links connecting more essential custom-
ers. The speaker described optimization models for these
situations and compuiiaional experience in solving large-
scale problems with hundreds of nodes.
The speakers from nine countries discussed diverse appli-
cations in fields such as engineering, computer science, op-
erations research, transportation, telecommunications,
mian ufactui ring, anId airline scheduling. Since researchers in
network optimization come from many different areas, the
conference provided a unique opportunity for the cross-
discipliinary exchange of recent research advances as well as
a foundation lor joint research cooperation and a stimulus
for future research. To give an idea of the topics discussed,
a few are briefly described below.
Anna Nagiurney (University of Massachusetts) discussed
"Massively v I Comnputation ofI)Dynamic'Trafffic Prob-
lems Modeled as Projected Dynamical Systems." Compu-
tational results on the CM-5 and I BM SP2 on several traffic
network examples were reported.
Warren Powell (Princeton) gave a talk on "Approximations
for Multistage Stochastic Networks" in which he discussed
recent results of his work which arose out of dynamic re-
source allocation problems.
Michi ael orian and I)enis Lcbetuf(Monrreal) presented an
efficient implementation of the network simplex method
which uses an extended Predecessor Index (XPI) data struc-
ture and a imetaheuristic for the choice of pivot.

Do(..I ,' I I, I Georgia' ech) and Cynthia Biarnhart (MIT)
presented their work on "Submodular Network Design
Problems." These problems ,. i I, involve opening a
subset of network elements (nodes, arcs or paths) from some
larger candidate set.
Dimitri Iitrtsekas (MITI) reported on recent algoritihmic and
implementation developments using a ( version of the
REIAX code and standard network I I I I ... I as
initialization techniques based on a recently proposed auc-
tion/sequential shortest path algorithm,.
Robert R. Meyer (University of Wisconsin-Madison) re-
ported on "Optimal Equi-Pairition: Billion Variable Qua-
dratic Assignment Problems." IHe presented an efficient
method for assigning the cells of a uniform grid among an
arbitrary number of processors so that load balancing con-
straints are observed while minimizing '' ,II perimeter
of the partition.
Michael Grigoriadis li analyzed the complexity of
fast approximation schemes f or problems charactciized bv
a number oFdisjoint convex blocksk) and a number of block-
separable nonnlegative convex (coupling constraints).
Michael I .. (University of Maryland) gave a talk on "The
Rural Postman vs. the Traveling Salesman: Modeling Prob-
lems on the Border Between Arc .... and Node Rout-
ing." He formulated several complex routing models based
on actual experience with the application areas of meterr
reading, mail delivery and refuse pickup.
Other invited speakers included: Karen Aardal, Ronald
Armstrong, John i Rainer Burkhard, Narsingh Deo,
Antonio Frangioni, Robert Freund, Steven Gabriel, Alexi
Gaivoronski, Jean-Louis Goffin, Andrew C . i Chi-
Gecun Han, M. Joborn, i,, .... m, Bruce Lamar,
P.O. I 11 Athanasios Migdalas, Michael Patriksson,
Rekha Pillai, I azaros Polymeniakos, Aubrey Poore, Motakuri
Ramana, PIh.L.Toint,Tleodore Trafalis,and GuoliangXuie.
Several participants were friomi industry' and national labo-
ratories including IBM Watson Research Center, ATI&T
Bell Laboratories, Argonne National Laboratory, NEC
Research institute, ORNL Oak Ridge, CAPS Logistics,
ITALTEL Italy, and ICF: Kaiser International.
Proceedings of the c .i I C t 11
be published later tih
-I'AN S M IA' AItM )O,


'51,1U FI P IEN 4 ACH1

N .

Harlan Mills

1920 1996
Harlan D. Mills died ai his resi-
dence in Vero Beach, Florida, oil
January 8, 1996, at the age oi 76.
Harlan received his Ph.D. in Math-
ematics at Iowa State. He served oni
rhe faculties of Iowa State, Princeton
University, New York University,
University of Maryland, University
of Florida, and was Professor of
Computer Science at the Florida In-
stiture of technology In recent
years, Harlan was recognized for his
brilliant work in software develop-
ment (chief programmeri teams, top-
down design, structiired program-
ming and cleanroom software engi-
neering), but he began his career
working in mathematical program-
ming and operations research. His
paper "Marginal Values of Matrix
Games and Linear Programs" (pp.
183-193 of Linear Inequcaciies and/
Related Sysemsi, H.W. iKuhn and
A.W. Tucker, eds., Princeton Uni-
versity Press, 1956) was one of the
first to investigate this area.
Fie was one of the founders and
President of the Princeron-based
consulting fiir Mathematica, Inc.
He worked for IBM and was anl
IBM Fellow and a member of its
Corporate Technical Committee, a
technical staff member of RCA and
GE, and President of Software Engi-
neering Technology. In 1986, he
was Chairman of the Computer Sci-
ence Panel for the U.S. Air Force
Scientific Advisory Board; from
1980-83, he was (ovcrnor of the
IiEE Computcer Society; and from
1974-77, he was Chairman of thei
NSF Computer Science Research
Panel on Software Methodology.
Harlani epitomized tilhe rare scientist

who knew w how to integrate the ideas
and methods of computer science,
mathematics and operations re-
search. All of us have been influ-
enced and have benefited from his
productive career.

Steven Vajda

1901 1995
Steven Vajda, one of mathematical
programming's truie pioneers, passed
away after a short illness oin Decem-
ber 10, 1995.
Born in Budapest in 1901, lie stud-
icd mathematics primarily in Vienna
with shorter visits paid to Berlin and
(Gottingen, obtaining degrees in ac-
tuarial science and mathematics. Af-
ter qualifying, he worked as an actu-
ary in Hungary, Romania and Aus-
tria. Inl 1939, just before the out-
break of World Wlar II, he moved to
England. Like many others arriving
from continental IEurope at that
time, Steven Vajda was interned fior
six months on the Isle of Man where
he taught mathematics and partici-
pated in establishing a "do-it-your-
self" university. During most of
World War II lie worked Ior an in-
surance company at Epsom utic in
1944 was invited to join the British
Admiralty as a statistician, soon ris-
ing to Assistant D)irector of Phyisical
Research and later of Operational
Research. Inl 1952 he was promoted
to Head of Mathlematcics (iroupi at
the Admiralty Riesearch Iahoraroriy
Pat R ivert was the first lProfessor of-
OR in the UK (Lancaster Univer-
sirty, 1963). Steven Vajda became tile
second when lie joined Birminghami
University in 1965, a position he
held until his retirement in 1908
when he became a FCellow., In 1967

lie was invited by Sussex University
to become a Fellow and iii 1973 be-
came Visiting Professor of Math-
ematics, in which role he continued
actively, teaching and writing re-
search papers, for about 22 years, a
record which is unsurpassed in tihe
UK and probably anywhere outside
the UK as well.
Vajda was awarded an honorary de-
gree (D.Tech. h.c.) by Brunel Uni-
vcrsity. His eminence was also recog-
nized biv the Opcrational Research
Society (ORS) in the award of its
Silver Medal, followed in 1995 by a
Debts to Steven Vajda are in one
wav or another owed by many. After
joining the Admiralty, he spent
about 50 years consciousiy or un-
consciously motivating the careers of
ILIinumeros OR) workers. He exerted
this influence directly by teaching
and conference presentations and in-
direcrly by his writings and by the
example ofhis life.
In appreciation, a group of friends
andi II , joined forces and sug-
gested to the Mathematical Pro-
grammingil Stuidy (Croup, ORS, that
a special imeting should be
organiscd tro celebrate his work as
tlie tr11 f'outndingi hi i er of math
ematical prograllmming in the U l\,
Focusing, on dualily, ihe meeting
was eventually held illn Lononn on

I I 11 Among the high-
Sard oif thei (Com-
. "IS ciisand tile warm
I I I. I afterwards at the
S..i . I I. IIIP. ence of m them ari-
,I ... ......... was acknowledged
and an early volume (Vaida, 1956)
was recalled as the verv first book in
Europe on linear programming, be-
ing translated into French, German,
Japanese, and Russian. It is indeed
Steven Vajda who could rightly
claim to have inrodticed the subject
to both Europe and Asia. A report
on the festive 10 February meeting
appeared shortly after (Simons,
1995). Anoteric visible outcome is
the forthcomning Speciia/l i:iioi of
ljouri/l ofi Mah,'maics ill Bsiness
and Industry edited by S. Powell and
1T.P. Williams ( I and Will-
ia is, 1996).
Vajda's fifteenth book, A Mat/h-
eniiiticil Kialeidoscope, co-authored
bv F.meritus Professor Brian
( I came our just a few weeks
before his death. The biographical
data above is based on the section
"About our Authors'" Iound therein,
on conversations with Professor B.
(C and L.B. Kovacs, on an in-
terview in OR Newsletter (B father,
1995), and on tile citation prepared
by Professor Maurice Shutler for tiie
Award of the Companionship of- die
(Oprational Research Societi y to
Steven Vajda (Shutler, 1995).
ii is a gift of grace to enjov a long
life without the physical
horrors of old age and even more so
to preserve both a warm heart andl a
brilliant mind to tihe end. Those
gifts were granted to Professor
Steven Vajda.

I\( rarup@)B I'
krarup@dl, ,-, .11


N' 49

PA(;I 12

MARCH 1996



Bather, )., "An interview with
Steven Vajda," OR Newsletter,
January 1995.
C(ioollyvii, B.nd S. Vaida, A Malh-
cnalical Kaeidoscope, Albion Pub-
lishing, Chichester, 199.
Kiarup, ]., and S. Vajda (1996a),
"On IToMicclli's (geometrical solution
to a problem of Fe crat," to an|ppeai
in (Powell and Williams. 1996).
Krarup, J., and S. Vajda ( 1996b),
"Visualizing dualitn," submitted ior
Powell, S., and H.P. Williams, IEds.
( 1996), Speciil Editioni of/jolrn/ of
Muthemantii's m Busi'ss and I!dustr':
Dulny:/ In ,clebratiou of iork o /'.

Shutlcr, M.F. (1995), "Steven
Vajda," citation for the (Companion-
ship of O)RS, to appear in (Powell
and Williams, 1996).
Simon, R., i I,, father of Brit-
ish LP is honottured. Celebration of
Stcven Vajda's work," OR Newslet-
ter, April 1995.
Vajda, S., Ylcory of (,n6Ics and ,in--
(,(r Methucn. 1956.
Vajda, S., Malhenmati(all ,
r\ing, Addison-Wcesity, 1961.

Svata Poljak

Svatopluk Poljak died on April 2,
1995, at age 44 in ani auto accident
near I I He is survived by his
wife Jana and theit two sons Honza
and Virek.
I le was born il Plague on October
9, 1951, and did his training at
Charles University in Prague. IH re-
ceived his RNDr diploma (doctorate
ill natural sciences) in 1976 and ob-
rained his Phi) in 1980 under the
supervision of Zdenck Hcdrlino.
Svata taught at the Czech Technical
University iln c i from 1979 to
1986, when ihe rejoined Charles
University as a senior researcher. He
was awarded a qualification degree
from the Czechoslovak Academv of
Sciences in 1990. In April 1994
Svata moved to the Universirt of
Passaut to take up a position on tihe
aculity of Malothematics and Com-
ptieIr Science.
cvaca's contributions iare best dem-
onstrated by over 95 publications in
diverse areas of combinatorics and
discrete optimization. At the time of
his death he was working intensively
on solving discrete optimization
problems such as the max-cut and
stable set pIroblCems, using eigenvalueC
techniques and nonlinear program-
miing approaches. The fasr approxi-
marion algoriithms for finding a
maximum cut in a graph that re-

ccntly came in the spotlight after the
breakthrough paper by Michel
Gonemans and David Williamson,
find their root in work ofSvata
(with (Charles Delorme) on eigen-
value methods for graph problems.
Indeed, the two approaches ofi
Michel and David and of Charlies
alld Svaia are, illn ct, dual in theo
sense of semnidefilite progranmmiing
quality. Svata and Charles conjec-
tured in 1993 thar the bound pro-
vildd by this approximation is very
close to thle true optimum (within
13 percent); C(;oemllns and Wiliams
succeeded in proving an estimation
slightly larger than the conjecturled
one. These results are very interest-
ing from a theoretical point of view;
moreover, They are promising for ilhe
practical purpose of solving max-curt
problems, as these can now be Tack-
led via interior point methods.
It would be difficub to summarize
all of the significall contributions
Svaia made. One area where his

work had a bigi impact, though per-
haps not so well-known, was in
neiiral networks. He (along with
Dan Turzik) found anl elegant solu-
tion to an open question concerning
the periodical bChaviour of finite au-

Svara collaborated with mian y people
throughout the world, always bring-
ing warmth and friendship in work
relationships. Many of us liave lost
in him a precious collaborator and a
dear friend. We all miss him verv
11 much.
-1i10 Nl{ tl I AIIltlN I

A list of Svata's publications as well
as several of his publications have
been posted by I lcnrv Wolkowic/
and can be found on tihe
WWW > at URL:\\>


The council has decided on the locations and
the chairs of the Organizing and Program
Committees of the next two IPCO-meetings.
They are as follows:

IPCO 6, 1998

HIouston, TX, U.S.A.
Chair of the Program Commnittee: Robert i. Bixbv
Chair of the Organizing Committne: 1. Anlldrew Boyd

IPCO 7, 1999
Craz, Austria
Chair of the Program Committee: (erard (Cornujols
(Chair ofdthe Organizing Committee: Rainer 1. Blurkard

N" 49

MARCH 1990(


N" 49

MARCH 1996

S~ ~ l~


Ilrl(L)1)111 tranlll Al)lInIUxa~iI ~II alilinh

A n////i/i,/' Programming

By O.L. Mangasarian
Classics in Applied Mathematics 10,
Philadelphia, 1994
ISlN 0-8)876 Ml1-x

Nmuneirical optimization (noniinear
programming) is a rich and practical
subfield of applied mathematics.
Undeirpinning1 the mitethods of (cil-
t l strained) numerical opI imz1 I oII is a111
clegan th eo ry connecringconi vexity, optinmaliity conditions,
duality, and various aspcts o niioniilealrity. A serious stu-
dentI 01r us ofopti izall ti should study tl is rheor v and
have the important concepts readily available. Thle best
source over the last two decades, inl terms ofa healthy mix
of' rigor, ibevity, and accessibilirty, has bcecn O( vi
Mangasarian's," -appily.SIAM has
chosen to reprini this book in its Classics series.
Si .. i ic .tie memerof myl
bookshelf lor ovcl I vcyars. (On occasions too 1 erlilcous
to coIunt, I have ireaiced ifr my slim blue copy to review
a convexity resuIII or r check a topological definition. I
amn usually rewarded, because this book has a remarkable
quality': the important theoretical concepts oficonstrai ned
oprimization are there, easy to find, easy to undersutandl
Si Chaprters 1 -4 with basic
definitions, a discussion ofi theorems of the alternative"
and an hintoducCon to the thitndamelcals of convexitv.
Chapter 5 is concerned with optinralitly condition with-

out assuming cdii'ferc ia bility. li I lls chapter Iir iwo
s'rasons. hilst, tie mi aleria is im ipo 'lant but not I icom ll ontl
presented (typically. arleast I i . i ..
Second, the author does an excellent job of cleariv Indi-
caring when convCxiIy is required and when it is o.

Chapter l 6 discuss diiffereniiabil convex and concave
functions; Chapter 7 presents well-used optinmality co
editions; Chapter 8 disciusss dt uialtv (a topic itha has played
a prominent aIgoritihllic loic it icccnd years) Tihe pde-
scniation here is distinguished by simple diagrams to
il ii ii ' Vil ilII Oous 1coInsIr1111 )i t lliiic'ations

liet relutiolships a.ilongist Various pCrobllls/opltin aiit

pseudoconvex atmnotions are dis (10. TEhe hallmark oi hcese chapters is hat subile reauion-

ships are exposed in simple diagramniaiic fornl.
Chapter I 1 addslion il neaqxl '/ lconsaits 1 the mix.
)ptimality and idualit results ai e 're-C xa mine d in tls nieiw
light, Fourl appexndices, I (ic chapter I i form I all
important part of this book: Relevant topics from ilnmca!
algebra, topology, and real analysis al summary ed.i These
appendices comain important background material for
lihe main part of rhe book, Moreover, they represent a
convenient packaging of hasic mnatheimatics used it op-
timi/l 1111on.
in conclusion, VNoni'near indeed a cis-
sic, I, I i the auit or, "It is a concisC, igorous,
yet accessible accolucll of futndalmentals of constrained
optnmI.ation theory that is t usefiu both to he beginning
student asweli as their acive resIcheIr." This is a t resource
tliat can benefit every serious "student" ofoptimi.xzaiton.
It i ims portent o Iunlldrsttaind whia a book is noi, as weil
ats what i is. Tiis book is not ab)ut algorihmt or mth-
ods (none arle discussed). It is cexrailinl not about prac-
rical computing issues. is not concerned withlx complexity
issues, It is also difficult ior ine to imagine this book as
a primary text f orla course, except perhaps a course caught
by an expert inll this area who could "smooch out" the
material with some exampcles of applicability and addi-
tional motivating material. The value of this book is as
a resource: it is a wonderful sum Vary of important sup-
porting theory or nonlinear optin zarion, especially with
respect to constraints.
mIot -\IA I (I I \t \\

I 1, I i

Ns a i9 ARH I99

Control and Optimization

by 1B.D. Craven
Chapman and Halli
LIondon, 1995
I ('N 0- Il -589/)0 t

I lhis volume tirears control problems
governed by ordindiaY diiiLereCial and
difference equations from a unified
standpoin oi optimiz.aion il normled
vecior spaces. Here aire a vast numbelltC.'
of, articles in scientific journals on this ropic, but no
comprehensive monograph has previously appea)cred.
]ThelefTore, this bool< is a hrsI t step in this direction,
The ideas of opUnmiation are inhi oducect il tihCe isi
chapet bli m1eanlls oi some simpic cxamplels iN finiie and
infinite dimensions. Some mathcma Ical background is
also presented spanning such diverse material as simple
matrix calculations and tlhc rather deep idea of wealk
compact nessand Alaoglu's theorem. Basic iicasule tieorly
is lllmmariliZ(CeI il 011 o page.
(ihaptcer'l wode scribecssix/ iasicdynamic conroi models
and discusses die respective cost (inuctions lo b optimized:
adverising models, investment models, ploducuion and
iIvemory models, water management models, ish popu-

I ( ionia COlI ((l illCo-lhasr(I 1 b e S ill 5011 iclimlj spacI (

a CCii All i ( is IN ioniN ( i iIi ri a INa IIiIi/ ciel l cixpId e ol h isiO

ChapletRNI ((10(1 gives a hA baane intillucdiok to honvx
1110 (Ioal iNNI B N11( 111 N3116 /.A11 haC ibsric( Spacs. Top

a pics B iesci luonl l 'iii (tic ii Lifo ((IC o ii c LIlaivclo
(~IN T JI l ((((I /A f(( II INC i~Ii CIS N (IUiAINLI II NINAI N.I~C Iol

conditions for conriol problems are derivedilr in Chapletnii
Fo1 ul N ii, both ifio idlei lland Continu u I tt N 11 ( I Aoblel

The '(A l (I ((INN11 (I I th d.. IMIXI I ono ( II I ( 11161 IC ,

aNh detaie prooI (( w1(1( 1 (isN( de layd nil( Chapel ((l' l(lC

theIcc le IIIIiNC (((I not fi l any m(aLicii il secon NC oli (( L
Chap Iter F iive (lIa i ((i i o i t ire Coll I CN111111I's IntIio

dUik C I Ii ( INC Onl(AC W 'IhNIICCIIN Oi lil iN a iii ail l iNiOI
(It(li. l C as V1r tie oi o ap e l lie lyIIrl m d l

Chapter Six is dedicated to algorithms fior optimal con-
trol problems. TIhere have been some micilestming exten-
sions of finite dimensional optimization algoli ihms
(Newton-, quasi-Ncvwton- a.nd conjLIugate gradient algo-
rithmsN) to infiite diciiensioiln blui tilCiCse agoli hiis aie
neirhciil A'ficien( in theircoi mputational raIiz ion norare
theV inmplciimented in robus pirotiessiona!-like packages.
(The state oF r i ncii compltatioinal opitiml control,
dirct and indirect meto ds, shooting, finite difficence
and collocation techniques, D whc arc thie nCaroal ap-
proacihs i or diherlenial equations as side conisu aiNs, is
sunm arized in I 11.)
Except lor ithe many miisprints in mathematical i-ormu-
law, where vectors are nor ,yped in boldticc, which greasy
hinders ithei rceadabilivo t'the ex dthis hookcan beCwarmi

recome'nici'idced to matahemiaticians Vwho ate interested in
a quick inliroduction to the subject. It is not equally
worthwhile lor engineers inierested in a more ]lcuristl(
I1II Burlirsch, R., D. Kraft: Computational
Optimal Control. Birkhauser, Boston, 1994.
-)IE1:1 i KRAI I

[ wish to enroll as a member of the Society.

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

I will pay my membership dues on receipt ofyour invoice.

I wish to pay by creditcard (Master/Euro or Visa).

NiUMl1 ItI.XI RY i I'I-

,11\11 IM N A k Id

Mail to:
The Mathematical Programming Society, Inc.
c/o International Statistical Institute
428 Prinses Beatrixlaan
2270 AZ Voorburg
The Netherlands

Cheques or money orders should be made payable to
The Mathematical Programming Society, Inc., in
one of the currencies listed below.
D)ues for 1996, including subscription to the journal
Mathematical Progriamming, are I ). 105.00 (or
$60.00 or DM94.00 or 39.00 or I I 'i or
Student applications: Dues are one-half rhe above
rates. Have a faculty member verify vourC student sta-
Lus and send application with dues to above address.
Faculty verifyingl staLus

istii ntion

N" 49


Donald W. Hearn, I *ilIOn
Karen Aardal, I'ATIUliI-S 1ii)1iOR
Utrecht University
Department of Computer Science
PI.O. Box 80089
3508 TB Utrecht
The Netherlands
Faiz AI-Khhayyal, soF rw,w & <( OWMl[I\I [I o, 1
Geonrgia Techl
Industrial and Systems Enginieering
Atlanta, GA 30332-0205
Dolf Talman, BOOK RIVII-W HIi,1I
Department of Econometrics
Tilburg Universitv
P.O. Box 90153
5000 LE Tilburg
The Netherlands N
Elsa Drake, I-Si(,mNi-
(;AT(O]REngineering PUIi ICA'iI ION sl.:viKcs

Jouriml colllnens are' subject to chan rl by fIlc pub,, I, b


MARCH 1 '9)(6

emember that

preregLisr.tion for

ISNMPL- is vi,

World V-iide \o.b:

http://dmawww epfl ch/



The t-mai is:

ismp97L'masg 1.epfl.ch

4)[eadlinc f6., rhc

nex O')TIMA

i, X av I *. I),)(o.

0 P T I M A


Center for Applied Optimization
371 Weil Hall
PO Box 116595
Gainesville FL 32611-6595 USA


University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs