Handbook of Global Optimization: Volume 2 PDF
Preview Handbook of Global Optimization: Volume 2
Handbook 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 in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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-
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