Table Of ContentHandbook of Global Optimization
Volume 2
Nonconvex Optimization and Its Applications
Volume 62
Managing Editor:
Panos Pardalos
Advisory Board:
J.R. Birge
Northwestern University, U.S.A.
Ding-Zhu Du
University of Minnesota, U.S.A.
C. A. Floudas
Princeton University, U.S.A.
J.Mockus
Lithuanian Academy of Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
G. Stavroulakis
Technical University Braunschweig, Germany
The titles published in this series are listed at the end of this volume.
Handbook of Global
Optimization
Volume 2
Edited by
Panos M. Pardalos
Department of Industrial and Systems Engineering,
University of Florida, Gainesville, FL, U.S.A.
and
H. Edwin Romeijn
Department of Industrial and Systems Engineering,
University of Florida, Gainesville, FL, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-5221-9 ISBN 978-1-4757-5362-2 (eBook)
DOI 10.1007/978-1-4757-5362-2
Printed on acid-free paper
All Rights Reserved
© 2002 Springer Science+B usiness Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1st edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Contents
Preface vii
1
Tight relaxations for nonconvex optimization problems using the 1
Reformulation-LinearizationjConvexification Technique (RLT)
Hanif D. Sherali
2
Exact algorithms for global optimization of mixed-integer nonlinear 65
programs
Mohit Tawarmalani, Nikolaos V. Sahinidis
3
Algorithms for global optimization and discrete problems based on 87
methods for local optimization
Walter Murray, Kien-Ming Ng
4
An introduction to dynamical search 115
Luc Pronzato, Henry P. Wynn, Anatoly A. Zhigljavsky
5
Two-phase methods for global optimization 151
Fabio Schoen
6
Simulated annealing algorithms for continuous global optimization 179
Marco Locatelli
7
Stochastic Adaptive Search 231
Graham R. Wood, Zelda B. Zabinsky
8
Implementation of Stochastic Adaptive Search with Hit-and-Run 251
as a generator
Zelda B. Zabinsky, Graham R. Wood
v
vi HANDBOOK OF GLOBAL OPTIMIZATION VOLUME 2
9
Genetic algorithms 275
James E. Smith
10
Dataflow learning in coupled lattices: an application to artificial 363
neural networks
Jose C. Principe, Curt Lefebvre, Craig L. Fancourt
11
Taboo Search: an approach to the multiple-minima problem for 387
continuous functions
Djurdje Cvijovic, Jacek Klinowski
12
Recent advances in the direct methods of X-ray crystallography 407
Herbert A. Hauptman
13
Deformation methods of global optimization in chemistry and phy- 461
sics
Lucjan Piela
14
Experimental analysis of algorithms 489
Catherine C. McGeoch
15
Global optimization: software, test problems, and applications 515
Janos D. Pinter
Preface
In 1995 the Handbook of Global Optimization (edited by Reiner Horst
and Panos Pardalos) was published. This handbook was very well re
ceived by the optimization community. However, may topics that are
used by practioners were not included in the 1995 handbook. We decided
to edit a second volume with the hope to fill this gap. Together, the two
volumes of the handbook cover a more complete and broad spectrum of
approaches for dealing with global optimization problems.
Many large-scale optimization problems encountered in practice can
not be solved to optimality using traditional optimization techniques.
Over the last decade, a large variety of heuristic techniques been pro
posed for specific optimization problems. Etymologically, the word "heu
ristic" comes from the Greek heuriskein (to find). Recall the famous
"Eureka, Eureka!" (I have found it! I have found it!) by Archimedes
(287-212 B.C.). Heuristics play a key role in the solution of large un
structured global optimization problems. As H. Wilf writes in his 1986
book "Algorithms and Complexity" about heuristics: ". .. methods that
seem to work well in practice, for reasons nobody understands ... ". The
application areas in which heuristics can and have been successfully ap
plied include machine and crew scheduling problems, vehicle routing
problems, telecommunications, supply chain management, and finance.
This second volume of the handbook of global optimization is com
prised of articles written by the experts in the fields dealing with modern
approaches to global optimization, including different types of heuristics.
Our goal was to provide a true handbook that does not focus on partic
ular applications of the heuristics and algorithms, but rather describes
the state of the art for the different methodologies. Topics covered in the
handbook include various metaheuristics, such as simulated annealing,
genetic algorithms, neural networks, taboo search, shake-and-bake meth
ods, and deformation methods. In addition, the book contains chapters
on new exact stochastic and deterministic approaches to continuous and
mixed-integer global optimization, such as stochastic adaptive search,
two-phase methods, branch-and-bound methods with new relaxation
vii
viii HANDBOOK OF GLOBAL OPTIMIZATION VOLUME 2
and branching strategies, algorithms based on local optimization, and
dynamical search. Finally, the book contains chapters on experimental
analysis of algorithms and software, test problems, and applications.
The target audience of the handbook is graduate students in engineer
ing and operations research, academic researchers, as well as practition
ers, who can tailor the general approaches described in the handbook to
their specific needs and applications.
We would like to take the opportunity to thank the authors of the
chapters, the anonymous referees, and Kluwer Academic Publishers for
making the publication of this volume possible.
PANOS PARDALOS, EDWIN ROMEIJN
Chapter 1
TIGHT RELAXATIONS FOR NONCONVEX
OPTIMIZATION PROBLEMS USING THE
REFORMULATION-LINEARIZATION/
CONVEXIFICATION TECHNIQUE (RLT)
Hanif D. Sherali
Grado Department of Industrial and Systems Engineering {0118}
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061
[email protected]
Abstract This paper provides an expository discussion on applying the Reformula
tion-Linearization/Convexification Technique (RLT) to design exact
and approximate methods for solving non convex optimization problems.
While the main focus here is on continuous nonconvex programs, we
demonstrate that this approach provides a unifying framework that can
accommodate discrete problems such as linear zero-one mixed-integer
programming problems and general bounded-variable integer programs
as well. The basic RLT approach is designed to solve polynomial pro
gramming problems having nonconvex polynomial objective functions
and constraints. The principal RLT construct for such problems is to
first reformulate the problem by adding a suitable set of polynomial
constraints and then to linearize this resulting problem through a vari
able substitution process in order to produce a tight higher dimensional
linear programming relaxation. Various additional classes of valid in
equalities, filtered through a constraint selection scheme or a separation
routine, are proposed to further tighten the developed relaxation while
keeping it manageable in size. This relaxation can be embedded in a
suitable branch-and-bound scheme in order to solve the original prob
lem to global optimality via a sequence of linear programming problems.
Because of the tight convexification effect of this technique, a limited
enumeration or even the use of a local search method applied to the
initial relaxation's solution usually serves as an effective heuristic strat
egy. We also discuss extensions of this approach to handle polynomial
programs having rational exponents, as well as general factorable pro-
P.M. Pardalos and H.E. Romeijn (eds.), Handbook of Global Optimization, Volume 2, 1-63.
© 2002 Kluwer Academic Publishers.
2 HANDBOOK OF GLOBAL OPTIMIZATION VOLUME 2
gramming problems, and we provide recommendations for using RLT to
solve even more general unstructured classes of nonconvex programming
problems. Some special case applications to solve squared-Euclidean,
Euclidean, or fp-distance location-allocation problems are discussed to
illustrate the RLT methodology.
1. Introduction
In this paper, we provide an expository discussion on the Reformula
tion-Linearization/Convexification Technique (RLT) for solving (exactly
or approximately) wide classes of nonconvex optimization problems. The
basic concept for this approach was developed for polynomial program
ming problems having nonconvex polynomial objective and constraint
functions. However, as we shall see, the technique can be adapted along
with supporting approximation procedures to solve a variety of more
general nonconvex optimization problems. Furthermore, this technique
provides a unifying framework that can accommodate discrete problems
such as linear zero-one mixed-integer programming problems and general
bounded-variable integer programs as well. Indeed, a zero-one restric
tion on some binary variable Xj can be represented by the polynomial
(continuous) constraints x J ( 1 - x j) = 0, where 0 :::; x J :::; 1. Furthermore,
if Xj can take on several general discrete values (which may or may not
be integral) in a set Sj = {OjkJ k = l, ... ,nj}, then we can represent
this via the polynomial constraint IJ:~1 (xJ - eJk) = 0. This viewpoint
permits the specialization of RLT techniques developed for polynomial
programs to such classes of discrete problems, but also reveals insights in
the reverse direction, where particular results or procedures developed
for the latter types of problems could be translated to the realm of the
more general class of polynomial programs.
The fundamental RLT approach operates in two phases. In the Re
formulation Phase, certain types of additional implied polynomial con
straints, are appended to the problem. The resulting problem is subse
quently linearized, except that certain convex constraints are sometimes
retained in particular special cases. This step constitutes the Lineariza
tion/Convexification Phase and involves the definition of suitable new
variables to replace each distinct variable-product term. The result
ing higher dimensional representation yields a linear (or convex) pro
gramming relaxation. In fact, by employing higher-order polynomial
constraints in the reformulation phase, this technique can be applied
to generate a hierarchy of such relaxations. A noteworthy feature is
that in several applications in the context of location-allocation, net
work design, economics, and engineering design, it has been demon-
