Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction

505 Pages·2000·9.706 MB·English

by  John Hooker(auth.)

L·gic-Based Methods
for Optimization
WILEY-INTERSCIENCE
SERIES IN DISCRETE MATHEMATICS AND OPTIMIZATION
ADVISORY EDITORS
RONALD L. GRAHAM
AT & T Laboratories, Florham Park, New Jersey, U.S.A.
JAN KAREL LENSTRA
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven, The Netherlands
A complete list of titles in this series appears at the end of this volume.
Logic-Based Methods
for Optimization
Combining Optimization and
Constraint Satisfaction
John Hooker
A Wiley-lnterscience Publication
JOHN WILEY & SONS, INC.
New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
This text is printed on acid-free paper. ®
Copyright © 2000 by John Wiley & Sons, Inc. All rights reserved.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,
except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without
either the prior written permission of the Publisher, or authorization through payment of the
appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA
01923, [phone], fax [phone]. Requests to the Publisher for permission should be
addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,
NY 10158-0012, [phone], fax [phone], E-Mail: PERMREQ @ WILEY.COM.
For ordering and customer service, call 1-800-CALL-WILEY.
Library of Congress Cataloging in Publication Data:
Hooker, John, 1949-
Logic-based methods for optimization : combining optimization and constraint
satisfaction / John Hooker.
p. cm. — (Wiley series in discrete mathematics and optimization)
Includes bibliographical references and index.
ISBN 0-471-38521-2 (cloth : alk. paper)
1. Linear programming. 2. Mathematical optimization. 3. Logic, Symbolic and
mathematical. I. Title. II. Series.
T57.74.H66 2000
519.77—dc21 99-088732
10 9 8 7 6 5 4 3 21
To Peggy and J. T.
Preface
This book is for readers who wish to solve optimization problems more effec-
tively, or to integrate optimization and constraint satisfaction. It accomplishes
both tasks by analyzing and extending the role of logic in optimization.
The book is written for people with background in either optimization or
constraint satisfaction, but not necessarily both. For those new to constraint
satisfaction techniques, it contains three tutorial chapters on these and con-
straint programming. For those with limited background in optimization, it
provides examples and elementary explanations of the relevant optimization
methods.
The book is for practitioners as well as theorists. About two-thirds of the
book (Chapters 1-15) presents techniques and modeling frameworks that are
essentially ready for implementation. Some have already been successfully
implemented. The practitioner should therefore find the book of immedi-
ate value. For example, it is now possible to develop modeling and solution
software that combines optimization and constraint satisfaction methods in
a principled way. This book presents the main elements of the technology
necessary for its development.
The remainder of the book (Chapters 16-21) digs a little deeper. It suggests
unproven ideas that could require further development before application.
Nonetheless they may have the greatest potential for payoff.
The book is also suitable for a graduate seminar involving students trained
in optimization or constraint satisfaction/constraint programming. Earlier
drafts were used for such courses; students presented a selection of topics in
vii
viii PREFACE
class. The introductory chapter outlines some possible study plans. If the
view taken in this book is prescient, courses that fuse the two areas, and
textbooks that do the same, may eventually become the standard.
Throughout this project I have benefited from my work with many individ-
uals. Of these colleagues I will name only a few who have shared my research.
They are Kim Allan Andersen, Endre Boros, Vijay Chandru, Giorgio Gallo,
Omar Ghattas, Ignacio Grossmann, Peter Hammer, Gerald Thompson, and
V. Vinay. I have also collaborated with several present and former gradu-
ate students: Srinivas Bollapragada, Milind Dawande, Chawki Fedjki, Farid
Harche, Hak-Jin Kim, N. R. Natraj, Maria Auxilio Osorio, Greger Ottosson,
Gabriella Rago, Ramesh Raman, Erlendur Thorsteinsson, and Hong Yan.
I owe thanks to a number of persons for suggesting improvements and
spotting errors in the text, particularly, Omer Benli, John Chase, Oya Ekin-
Karasan, Richard Rosenthal, and Laurence Wolsey. I used earlier versions of
the book in courses at Carnegie Mellon and Bilkent universities, where my
students and colleagues help me clarify the message and suggested numerous
improvements in the manuscript.
JOHN HOOKER
Pittsburgh, USA
Contents
Preface vii
1 Introduction 1
1.1 Logic and Optimization 1
1.1.1 Optimization and Constraint Satisfaction 2
1.1.2 Constraint Programming 4
1.1.3 Development of Logic-Based Methods 6
1.1.4 Recent Applications and Software 8
1.2 Organization of the Book 9
1.2.1 How Much to Read 9
1.2.2 Background Material 11
1.2.3 A Practical Logic-Based System 12
1.2.4 A Deeper Analysis 12
2 Some Examples 15
2.1 Logic-Based Modeling 16
2.1.1 The Traveling Salesman Problem 17
2.1.2 The Assignment Problem 18
2.1.3 The Quadratic Assignment Problem 19
2.1.4 A J°b Shop Scheduling Problem 20
IX
X CONTENTS
2.2 A Knapsack Problem 23
2.2.1 An Integer Programming Model 23
2.2.2 An Integer Programming Solution 24
2.2.3 A Logic-Based Solution 27
2.3 Processing Network Design 31
2.3.1 An Integer Programming Approach 32
2.3.2 A Logic-Based Approach 33
2.4 Lot Sizing 37
2.4.1 An Integer Programming Model 38
2.4.2 A Logic-Based Model 39
3 The Logic of Propositions 43
3.1 The Idea of Propositional Logic 44
3.1.1 Formulas 44
3.1.2 Clauses 45
3.1.3 Conversion to Clausal Form 4^
3.1.4 Horn Clauses 48
3.1.5 Renamable Horn Clauses 50
3.2 Resolution 53
3.2.1 The Resolution Algorithm 53
3.2.2 Projection 55
3.2.3 Unit Resolution 57
3.2.4 Constraint-Based Search 59
4 The Logic of Discrete Variables 61
4.I Formulas of Discrete-Variable Logic 62
4-1.1 Formulas and Semantics 62
4.I.2 Multivalent Clauses 62
4-2 Multivalent Resolution 63
4.2.1 Full Resolution 63
4.2.2 Projection 65
4-2.3 Unit Resolution 65
4.2.4 Constraint Generation 66
4-3 Defined Predicates 67
5 The Logic of 0-1 Inequalities 69
5.1 Inequalities and Implication 70
5.2 Resolution for 0-1 Inequalities 73
CONTENTS XI
5.2.1 The Algorithm 73
5.2.2 Completeness of 0-1 Resolution 74
5.2.3 Resolution and Cutting Planes 76
5.3 Equivalent Inequalities 78
5.3.1 Characterizing an Equivalence Class 78
5.3.2 A Polar Approach to Checking
Equivalence 79
5.3.3 Polar Characterization of Equivalence
Classes 83
5.3.4 Canonical Inequalities 85
Cardinality Clauses 89
6.1 Resolution for Cardinality Clauses 90
6.1.1 The Classical Resolution Step 90
6.1.2 The Diagonal Summation Step 93
6.2 Generating Cardinality Clauses 95
6.2.1 Implied Cardinality Clauses 95
6.2.2 Generating Nonredundant Implications 97
6.2.3 Implied Contiguous Clauses 101
Classical Boolean Methods 105
7.1 Pseudoboolean Optimization 107
7.1.1 The Basic Method 108
7.1.2 The Basic Algorithm Revisited 110
7.2 Roof Duality 112
7.2.1 Roofs 112
7.2.2 The Roof Dual 114
7.3 Implied Constraints 116
7.3.1 Implications of a Linear 0-1 Inequality 117
7.3.2 Implications of a Nonlinear 0-1 Inequality 118
7.4 Matching Problems 120
Logic-Based Modeling 127
8.1 A Modeling Framework 128
8.1.1 The Basic Framework 129
8.1.2 A Growing Lexicon of Global Constraints 130
8.1.3 Element Constraints and Variable
Subscripts 131
8.1.4 Sum Constraints and Variable Index Sets 133