MARCH
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lMATH MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
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
Email:
hanifs@vt.edu
Warren P. Adams
Department of Math Sciences
Clemson University
Email:
wadams@clemson.clemson.edu
Introduction. Discrete and continuous nonconvex programming problems arise in a
host of practical applications in the context of production planning and control,
locationallocation, distribution, economics and game theory, process design, and
Design situations. Several recent advances have been made in the develop
ment of branchandcut algorithms for discrete optimization problems and in polyhe
dral outerapproximation 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 ReforiiiilationLiiinariiatiinTeichiiqii' (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 01 and mixed 01 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 01 polynomial programs in n 0I variables, we have ([II,
12]) developed an nlevel hierarchy, with the iith 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 mixedinteger zeroone 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
gallimaufiry
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 higherdimen
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 01
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 levelI
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
polynomialtime 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
secondlevel 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
leveln. By inductivelv progressing
along these steppingstone
formulations, we have studied
sorne novel 'prsifsitciici/ issues for
certain constrained and uncon
strained pseudoBoolean 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 pseudoBoolean
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 01
programming g problems. In
particular, the analysis here
reveals a class of 01 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
property.
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
sparsity.
Although the Reformulaltion
Linearization Technique was
originally designed to employ
factors involving iiroonti variables
in order to generate zeroone
(mixedinteger) polynomial
programming problems that are
subsequently relinearized, 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 01
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.)
N"49
MARCH 1996
N"49
0j
'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 levelone
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 01 polynomial
programs l[9, ene'ompassing and
generalizing all known persistent
formulations.
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.
Zeroorne quadratic programs
and the quadratic assignment
problem. The zeroone 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 levelone 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 levelone
relaxation and by incorporating
dualbased 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 920, 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
valuebased procedure produced a
lower hound of 2229, the optmrnum
value being 2570 for this problem.
Recently, Rese'nde c' al. (()Opc'iioni
RIi'iih, IW995) have been able to
solve the first level RIT relaxation
exactly for problems of size up to
30 using an interiorpoint 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 firstlevel 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 01 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
firstlevel 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 01 solution.
Continuous and discrete
bilinear j ... ,. 'ii ii
problems. lie wellknown
nonconvex, NPhard 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
problems.
An enhanced firstlevel 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 nchandbound 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,
firstlevel 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 crossproduct
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 01 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 mixedinteger
bilinear programming problems,
the RLTI process described above
for the latter class of problems was
enhanced by Incorporating
conllstraint factor crossproducts 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 34 times faster in terms
of tile overall effort required an
compared with the C'P.LX based
approach.
Continuous and discrete
locatoionallocationl prob
le ns. 1The RLT strategy has been
unsecl to derive very effective
algorithms for capacitated,
multifacility, locationallocation
problems that find applications in
service facility or warehouse
location, or manufacturing facility
flowshop 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 mixedilneger,
zeroone, bilinear programming
problem of the general ftrln
studied in I8. Wie specialized the
discussed levelone 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 510',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 24''i of
optinmahlt and that this procedure
significantly enhances the size (o
problems solvable by a branch
andbound 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
locationallocation problem in
which the iii capacitaed service
facilities are to be assigned in a
onetoonle fashion to some i
discrete sites itn order to serve the
i1 custontrs, Vwihere tihe cost per
unit flow is determmed by stme
general lacilitvcustomer separa
tion based penalty Iunction 1131
This problem also turns out to
have the structure ol a separably
constrained mixedinteger 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 909'/ of
optimality.
Indefinite quntdratic pro
granms and poyhnoimial
progranmming problems. In
I31 ], we have developed a global
optimization procedure for
linearly constrained indefinite/
concaveminimnization 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
branchandbound 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. BenTal 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 endpoint. 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 13) 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 mixedinteger 01
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
interiorpoint 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
DM19521398.
References Related to
Authors' RLT Research
Ill W. P. Adams, A. Billionnet, and
A. Sutter, "Unconstrained 01
Optimization and Lagrangean
Relaxation," Discrete Applied
Mathematics, Vol. 29, Nos. 23, pp.
131142, 1990.
121 W. P. Adams and P. M. Dearing,
"On the Equivalence Between Roof
Duality and Lagranian Duality for
Unconstrained 01 Quadratic
Programming Problems," Discrete
Applied Matheimatics, Vol. 48, No.
1, pp. 120, 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. 4375, 1994.
14] W. P. Adams and T. A. Johnson,
"An Exact Solution Strategy for the
Quadratic Assignment Problem
Using RLTBased 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 01
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 ZeroOne Quadratic
Programming Problems," Manage
ment Science, 32(10), pp. 12741290,
1986.
171 W. P. Adams and Li. D. Sherali,
"Linearization Strategies for a Class
of ZeroOne Mixed Integer
Programming Problems," Opera
tions; Research, 38(2), pp. 217226.
1990.
18] W. P. Adams and H. D. Sherali,
' I ..1 in r Bilinear Program
ming Problems," Matlhematical
Progrant inig, 59(3), pp. 279306,
1993.
191 J. B. Lassiter and W. P. Adams,
"Strategies for Converting Various
01 Polynomial Programs into
Persistent Mixed 01 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
ZeroOne Programming Problems,"
SIAM Journal on Discrete Math
ematics, 3(3), pp. 411430, 1990.
112] H. D. Sherali and W. P. Adams,
"A Hierarchy of Relaxations and
Convex Hull Characterizations for
MixedInteger ZeroOne Program
ming Problems," Discrete Applied
Mathematics, 52, pp. 83106, 1994.
1131 H. D. Sherali and W. P. Adams,
"A Decomposition ,.i ... ;iii o for a
Discrete LocationAllocation
Problem," Operations Research, pp.
878900, 1984.
114] H. D. Sherali and W. P. Adams,
"A ReformulationLinearization
Technique (RLT) for Solving
Discrete and Continuous
Nonconvex Programming Prob
lems," Vol. XIIA, pp. 6178,
Mathematics Toiday, special issue
on Recent Advanceis in Mattematni
cal P1ro'granmoiinrg, ed. O. Gupta,
1994.
1151 H. D. Sherali, W. P. Adams,
and P. Driscoll, "Exploiting Special
Structures in Constructing a
Hierarchy of Relaxations for 01
Mixed Integer Problems," Working
paper, Department of Industrial
and Systems Engineering, Virginia
Polytechnic Institute and State
University, I I I.. .... VA 24061
0118, 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(14), pp. 197214, 1990.
1171 H. D. Sherali and A.
Alameddine, "A New Reformula
tionLinearization Technique for
..i ,.. Bilinear Programming
Problems," Journal of Global
Optimization, 2, pp. 379410, 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. 343364, 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. 133153, 1995.
1251 1. D. Sherali and Y. Lee,
I l,. i Representations for Set
Partitioning Problems Via a
Reformu lationLinearization
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. 1926,
S995.
127] H. D. Sherali, S. Ramachandran
and S. Kim, "A Localization and
Reformulation Discrete Program
ming Approach for the Rectilinear
Distance LocationAllocation
Problem, "Discrete Applied
Matlhematics, 49(13), pp. 357378,
1994.
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
ReformulationLinearization
Technique," Journal of Global
Optimization, 2, pp. 101112, 1992.
1301 H. D. Sherali and C. I.
Tuncbilek, "A SquaredEuclidean
Distance LocationAllocation
Problem," Naval Research
Logistics, 39, 447469, 1992.
131 HI. D. Sherali and C. iH.
Tuncbilek, "A Reformulation
Convexification Approach for
Solving Nonconvex Quadratic
Programming Problems," Journal of
Global Optimization,, pp. 131,
1995.
1321 H. D. Sherali and C. H.
Tuncbilek, "Comparison of Two
ReformiulationLinearization
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
PrimalDual Conjugate
,i ,.i;. Algorithm for
Specially Structured Linear and
Convex Programming Problems,"
Applied Malthemalics and Optimi
zation, 20(S15), pp. 193221, 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 closedformi
representation of mixedinteger
program' m value functions."
S.E. Karisch and F. Rendl,
"Lower bounds for the quadratic
assignment problem via triangle
decompositions."
A.V. Goldberg and R. Kennedy,
"An efficient cost scaling
algoritihm for the assignment
problem."
J.V. Burke and M.C. Ferris, "A
GaussNewton method for
convex composite optimization."
R. Sridhar, "Superfluous
matrices in linear
complementarity."
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,+ 1matrices,
bicoloring and total dual
integrality."
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
equalities."
F.B. Shepherd, "Applying
Leihian's theorems to packing
problems."
G. Pritchard, G. Giirkan and
A.Y. Ozge, "A note on locally
Lipschitzian functions (Slorl
Communication)."
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 (2429 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: JeanPhilippe Vial
H E .1.. : i .... ..I Studies
University of Geneva
102, Bd Carl Vogt
CH1211 Geneva 4
Switzerland
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. HiriartUrruty (President)
B. Monjardet (VicePresident)
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
Email: lao@cictfr
I h I IRIAi i UR I( ; li
N" 49
MARCH 1996
N~' 49
S0 P T I
Conference
S5th SIAM Conference on
Optimization, British Columbia
May 2022, 1996.
S18th Symposium on Math
ematical Programming with
Data Perturbations
George Washington University
2324 May 1996
SIPCO V, Vancouver, British
Columbia, Canada
June 35, 1996
SFifth International Sympo
sium on Generalized Convexity
LuminyMarseille, France
June 1721, 1996
F 7th Stockholm
Optimization Days
Stockholm, Sweden
June 2425, 1996
I IFORS 96 14th Triennial
Conference, Vancouver
British Columbia, Canada
July 812, 1996
IRREGULAR 96
Santa Barbara, California
Aug. 1923, 1996
SInternational Conference on
Nonlinear Programming
Beijing, China
Sept. 25, 1996
SSymposium on Operations
Research (SOR96)
Technical University
Braunschweig, Germany
Sept. 46, 1996
SSecond International Sympo
sium on Operations Research
and its Applications
(ISORA '96)
Guilin, China
Dec. 1113, 1996
SXVI International
Symposium on Mathematical
Programming, Lausanne
Switzerland, Aug. 1997
con0f eHrencffB
7th Stockholm
Optimization Days
Stockholm, Sweden,
June 2425, 1996
We welcome theoretical, comIputa
rional and applied papers for the 7rt
Stockholm Optimization Days, a
twoday conference on optimization,
to be held at K'T1H (Royal Institute
of Technology) in Stoctkholm, Swe
den, June 2425, 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
transportation.
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 email) to:
optdays@math.kth.se
or by mail to:
Optimization Days
Division of O)pimiization a(nd
Systems 'Theory
K'lIH
S100 44 Stockhoil
Sweden
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).
K IS, I I R wANBI iR,
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 46, 1996
Section
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
Deadlines:
Apr 1, 1996
Apr 15, 1996
Chairperson
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 cmail (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 email and electronic networks. So, whenever possible, use email and
the Well for communication. IThe email address for all these contacts is:
sor96@tubs.de
Certain keywords on the subject linewill
trigger automatic responses, such as
Subject Reply1
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the email interface
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help returns information
manuscript concerning manuscripts
for the proceedings
To communicate via the Web, start at
URIL
http://moa.math.nat.tubs.de/sor96
and follow the respective links
(WWWforms).
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) 5313917550
Fax: +49(0)5313917559
sor96@tubs.de
bsiekA I Bmo.math.na bs.de
bussieck@moa.math.nat.tubs.de
MARCH 1996
Annals of Operations
Research
Special Issue on Parallel
Optimization
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 stateoftheart 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
egories:
* 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,
etc.
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
theo@crt.umontreal.ca
or0
Catherine Routcairol
Laboraroire PRISM
University de Versailles
45, Avenue des lErasUnis
78035 Versailles Cedex, FIance
(+33 1) 39 25 40 88
fax: (33 1) 39 25 40 57
Catherine.Roucairol@prism.uvsq.fr
Second International Sympo
sium on Operations Research
and its Applications
(ISORA '96)
Guilin, China
December 1113, 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
1113, 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,
Microcomputer software o OR
methods. Paper s on irealworld appli
cations will be especially appreci
ated.
I lie symposium is sponsored bhy tie
AsianPacific Operations Research
Center within the Association of
AsianPacific Operational Research
Societies (APORS) and Ch(inese
Academy of Sciences (CAS).
The symposium chair is Professor
XiangStu 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: 86102541689
isora@amath3.amt.ac.cn
o"
Program Commnittee Chair:
Prof. D)ingZhu I)u, Compulter
Science Department, University of
Minnesota, Minneapolis, MN
55455, U.S.A.
fax: 6126250572
dzd@cs.umn.edu
Submissions should be written in
English, at most ten pages, and in
clude thle email 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
Cameraready manttscript idule:
September 1, 1996
For information about program,
registration and local arrangements,
please contact:
Dr. X.D. Hu
Fax: 86102541689
ISORA@amath3.omt.ac.cn
or Prof. I)ingZhii Du
fax: 6126250572
dzd@cs.umn.edu
N"49
Sh rXlHBBfK
Il ,1F
HI
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
processing.
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.uszeged.hu/
~globopt
FTP: ftp.jate.uszeged.hu
in the directory /pub/math/
optimization/globopt
SIIlBo ( St NDI s
csendes@inf.uszeged.hu
http://www.inf.uszeged.hu/
"csendes/
Report on the Conference
Gainesville, Florida
inference on Network Optimi
ition was held at the (Center for Applied
Spitimization at the University ofiFlorida, Feb.
S14, 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 CM5 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 WisconsinMadison) re
ported on "Optimal EquiPairition: 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, JeanLouis 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
SpringerVerlag.
I'AN S M IA' AItM )O,
I
'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.
183193 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 Princeronbased
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
198083, he was (ovcrnor of the
IiEE Computcer Society; and from
197477, 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.
SAl I (;ASS
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 "doityour
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
Companionship.
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, coauthored
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
~I
References:
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 anppeai
in (Powell and Williams. 1996).
Krarup, J., and S. Vajda ( 1996b),
"Visualizing dualitn," submitted ior
publication,
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, AddisonWcesity, 1961.
Svata Poljak
19511995
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 maxcut 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 maxcurt
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 wellknown, 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
laurent@dmi.ens.fr
II N iYWO I KOVIC/
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:\\>
http://orion.uwaterloo.ca/
"hwolkowi/.preprints/authors.d/
poliak.d/
CO01/1il
The council has decided on the locations and
the chairs of the Organizing and Program
Committees of the next two IPCOmeetings.
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
N" 49
MARCH 1996
S~ ~ l~
IJI I&
Ilrl(L)1)111 tranlll Al)lInIUxa~iI ~II alilinh
A n////i/i,/' Programming
By O.L. Mangasarian
Classics in Applied Mathematics 10,
SIAM
Philadelphia, 1994
ISlN 08)876 Ml1x
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 wellused 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
characterizationis.
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 'reC 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 illColhasr(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
conlditions.
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, quasiNcvwton 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 iormu
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(
approach.
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
Sw.Fr.80.00).
Student applications: Dues are onehalf 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
MARCH I 996
Donald W. Hearn, I *ilIOn
hearneise.ufl.edu
Karen Aardal, I'ATIUliIS 1ii)1iOR
Utrecht University
Department of Computer Science
PI.O. Box 80089
3508 TB Utrecht
The Netherlands
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Faiz AIKhhayyal, soF rw,w & <( OWMl[I\I [I o, 1
Geonrgia Techl
Industrial and Systems Enginieering
Atlanta, GA 303320205
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Dolf Talman, BOOK RIVIIW HIi,1I
Department of Econometrics
Tilburg Universitv
P.O. Box 90153
5000 LE Tilburg
The Netherlands N
talman@kub.nl
Elsa Drake, ISi(,mNi
PUI B LIS II) IY TItI
MATI I MATICAI I'PR(XCRAMMING; SOCIl 'Y &
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UNIVERSITY OF FLORIDA
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