Large Scale Linear and Integer Optimization: A Unified Approach

736 Pages·1999·25.951 MB·English

by  Richard Kipp Martin (auth.)

LARGE SCALE LINEAR AND
INTEGER OPTIMIZATION:
A UNIFIED APPROACH
LARGE SCALE LINEAR AND
INTEGER OPTIMIZATION:
A UNIFIED APPROACH
Richard Kipp Martin
Graduate School of Business
Universify of Chicago
~.
"
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Martin, Richard Kipp.
Large scale linear and integer optimization : a united approach /
Richard Kipp Martin.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-46l3-7258-5 ISBN 978-1-4615-4975-8 (eBook)
DOI 10.1007/978-1-4615-4975-8
1. Linear programming. 2. Mathematical optimization. 1. Title.
T57.75.M375 1999
5 19.7'2--dc2 1 98-46062
CIP
Copyright © 1999 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1s t edition 1999
AII rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, record ing, or otherwise, without the prior written permis sion of the
publisher, Springer Science+Business Media, LLC .
Printed an acid-free paper.
This book is dedicated to my parents, Bruce and Phyllis Martin.
CONTENTS
Preface xv
Part I MOTIVATION 1
1 LINEAR AND INTEGER LINEAR
OPTIMIZATION 3
1.1 Introduction 3
1.2 Linear and Integer Linear Optimization 5
1.3 A Guided Tour of Applications 7
1.4 Special Structure 21
1.5 Linear and Integer Linear Programming Codes 25
1.6 Other Directions 28
1.7 Exercises 29
Part II THEORY 33
2 LINEAR SYSTEMS AND PROJECTION 35
2.1 Introduction 35
2.2 Projection for Equality Systems: Gaussian Elimination 36
2.3 Projection for Inequality Systems: Fourier-Motzkin Elimi-
nation 39
2.4 Applications of Projection 46
2.5 Theorems of the Alternative 49
2.6 Duality Theory 57
2.7 Complementary Slackness 61
2.8 Sensitivity Analysis 65
2.9 Conclusion 75
2.10 Exercises 75
Vlll LARGE SCALE LINEAR AND INTEGER OPTIMIZATION
3 LINEAR SYSTEMS AND INVERSE
PROJECTION 81
3.1 Introduction 81
3.2 Deleting Constraints by Adding Variables 81
3.3 Dual Relationships 91
3.4 Sensitivity Analysis 93
3.5 Conclusion 99
3.6 Homework Exercises 100
4 INTEGER LINEAR SYSTEMS:
PROJECTION AND INVERSE PROJECTION ]03
4.1 Introduction 103
4.2 Background Material 105
4.3 Solving A System of Congruence Equations 114
4.4 Integer Linear Equalities 122
4.5 Integer Linear Inequalities: Projection 124
4.6 Integer Linear Inequalities: Inverse Projection 127
4.7 Conclusion 136
4.8 Exercises 137
Part III ALGORITHMS 141
5 THE SIMPLEX ALGORITHM 143
5.1 Introduction 143
5.2 Motivation 143
5.3 Pivoting 147
5.4 Revised Simplex 154
5.5 Product Form of the Inverse 162
5.6 Degeneracy and Cycling 168
5.7 Complexity of the Simplex Algorithm 177
5.8 Conclusion 178
5.9 Exercises 178
6 MORE ON SIMPLEX 183
6.1 Introduction 183
6.2 Sensitivity Analysis 184
6.3 The Dual Simplex Algorithm 191
Contents ix
6.4 Simple Upper Bounds and Special Structure 198
6.5 Finding a Starting Basis 201
6.6 Pivot Column Selection 205
6.7 Other Computational Issues 209
6.8 Conclusion 215
6.9 Exercises 216
7 INTERIOR POINT ALGORITHMS:
POLYHEDRAL TRANSFORMATIONS 219
7.1 Introduction 219
7.2 Projective Transformations 225
7.3 Karmarkar's Algorithm 231
7.4 Polynomial Termination 234
7.5 Purification, Standard Form and Sliding Objective 239
7.6 Affine Polyhedral Transformations 243
7.7 Geometry of the Least Squares Problem 254
7.8 Conclusion 258
7.9 Exercises 258
8 INTERIOR POINT ALGORITHMS:
BARRIER METHODS 261
8.1 Introduction 261
8.2 Primal Path Following 266
8.3 Dual Path Following 272
8.4 Primal-Dual Path Following 277
8.5 Polynomial Termination of Path Following Algorithms 283
8.6 Relation to Polyhedral Transformation Algorithms 292
8.7 Predictor-Corrector Algorithms 297
8.8 Other Issues 300
8.9 Conclusion 306
8.10 Exercises 310
9 INTEGER PROGRAMMING 313
9.1 Introduction 313
9.2 Modeling with Integer Variables 314
9.3 Branch-and-Bound 319
9.4 Node and Variable Selection 324
x LARGE SCALE LINEAR AND INTEGER OPTIMIZATION
9.5 More General Branching 328
9.6 Conclusion 341
9.7 Exercises 341
Part IV SOLVING LARGE SCALE PROBLEMS:
DECOMPOSITION METHODS 347
10 PROJECTION: BENDERS'
DECOMPOSITION 349
10.1 Introduction 349
10.2 The Benders' Algorithm 350
10.3 A Location Application 354
10.4 Dual Variable Selection 360
10.5 Conclusion 364
10.6 Exercises 365
11 INVERSE PROJECTION: DANTZIG-WOLFE
DECOMPOSITION 369
11.1 Introduction 369
11.2 Dantzig-Wolfe Decomposition 370
11.3 A Location Application 375
11.4 Taking Advantage of Block Angular Structure 384
11.5 Computational Issues 386
11.6 Conclusion 390
11.7 Exercises 391
12 LAGRANGIAN METHODS 393
12.1 Introduction 393
12.2 The Lagrangian Dual 394
12.3 Extension to Integer Programming 398
12.4 Properties of the Lagrangian Dual 402
12.5 Optimizing the Lagrangian Dual 408
12.6 Computational Issues 426
12.7 A Decomposition Algorithm for Integer Programming 429
12.8 Conclusion 434
12.9 Exercises 435
Contents xi
Part V SOLVING LARGE SCALE PROBLEMS:
USING SPECIAL STRUCTURE 437
13 SPARSE METHODS 439
13.1 Introduction 439
13.2 LU Decomposition 439
13.3 Sparse LU Update 446
13.4 Numeric Cholesky Factorization 462
13.5 Symbolic Cholesky Factorization 466
13.6 Storing Sparse Matrices 471
13.7 Programming Issues 472
13.8 Computational Results: Barrier versus Simplex 478
13.9 Conclusion 479
13.10 Exercises 480
14 NETWORK FLOW LINEAR PROGRAMS 481
14.1 Introduction 481
14.2 Totally Unimodular Linear Programs 482
14.3 Network Simplex Algorithm 493
14.4 Important Network Flow Problems 505
14.5 Almost Network Problems 513
14.6 Integer Polyhedra 515
14.7 Conclusion 523
14.8 Exercises 524
15 LARGE INTEGER PROGRAMS:
PREPROCESSING AND CUTTING PLANES 527
15.1 Formulation Principles and Techniques 527
15.2 Preprocessing 533
15.3 Cutting Planes 542
15.4 Branch-and-Cut 555
15.5 Lifting 557
15.6 Lagrangian Cuts 560
15.7 Integer Programming Test Problems 561
15.8 Conclusion 562
15.9 Exercises 563
xii LARGE SCALE LINEAR AND INTEGER OPTIMIZATION
16 LARGE INTEGER PROGRAMS:
PROJECTION AND INVERSE PROJECTION 565
16.1 Introduction 565
16.2 Auxiliary Variable Methods 573
16.3 A Projection Theorem 601
16.4 Branch-and-Price 601
16.5 Projection of Extended Formulations: Benders' Decompo-
sition Revisited 613
16.6 Conclusion 630
16.7 Exercises 630
Part VI APPENDIX 633
A POLYHEDRAL THEORY
635
A.1 Introduction 635
A.2 Concepts and Definitions 635
A.3 Faces of Polyhedra 640
A.4 Finite Basis Theorems 645
A.5 Inner Products, Subspaces and Orthogonal Subspaces 651
A.6 Exercises 653
B COMPLEXITY THEORY 657
B.1 Introduction 657
B.2 Solution Sizes 660
B.3 The Turing Machine 661
B.4 Complexity Classes 663
B.5 Satisfiability 667
B.6 NP-Completeness 669
B.7 Complexity of Gaussian Elimination 670
B.8 Exercises 674
C BASIC GRAPH THEORY 677
D SOFTWARE AND TEST PROBLEMS 681
E NOTATION 683