Table Of Content
Linear Programming:
Mathematics, Theory and Algorithms
Applied Optimization
Volume 2
The titles published in this series are listed at the end of this volume.
Linear Programming:
Mathematics, Theory
and Algorithms
by
Michael 1. Panik
University ofH artford
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
Panik. Michael J.
Linear programming mathematics. theory and algorithms / by
Michael J. Panik.
p. cm. -- (Applied optimization; vol. 2)
Inc I udes bib Ii ograph i ca I references and index.
ISBN-13: 978-1-4613-3436-1 e-ISBN-13: 978-4613-3434-7
001: 10.1007/978-4613-3434-7
1. Linear programming. I. Title. II. Series.
T57.74.P345 1996
519.7·2--dc20 95-41545
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell. MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
© 1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
In Memory of
Alice E. Bourneuf
and
Ann F. Friedlaender
TABLE OF CONTENTS
Chapter 1. INTRODUCTION AND OVERVIEW 1
Chapter 2. PRELIMINARY MATHEMATICS 9
2.1 Vectors in Rn 9
2.2 Rank and Linear Transformations 13
2.3 The Solution Set of a System of Simultaneous Linear
Equations 18
2.4 Orthogonal Projections and Least Squares Solutions 23
2.5 Point-Set Theory: Topological Properties of Rn 26
2.6 Hyperplanes and Half-Planes (-Spaces) 28
2.7 Convex Sets 31
2.8 Existence of Separating and Supporting Hyperplanes 34
2.9 Convex Cones 39
2.10 Basic Solut.ions to Lineal' Equalities 45
2.11 Faces of Polyhedral Convex Sets: Extreme Points,
Facets, and Edges 53
2.12 Extreme Point Representation for Polyhedral Convex
Sets 56
2.13 Directions for Polyhedral Convex Sets 60
2.14 Combined Extreme Point and Extreme Direction
Representation for Polyhedral Convex Sets 62
2.15 Resolution of Convex Polyhedra 63
2.16 Simplexes 64
2.18 Linear Fundionals 66
Chapter 3. INTRODUCTION TO UNEAR PROGRAMMING 69
3.1 The Canonical Form of a Linear Programming
Problem 69
3.2 A Graphical Solution to the Lineal' Programming
Problem 70
3.3 The Standard Form of a Linear Programming
Problem 72
3.4 Properties of the Feasible Region 75
3.5 Existence and Location of Finite Optimal Solutions 76
3.6 Basic Feasible and Extreme Point Solutions to the
Linear Programming Problem 81
3.7 Solutions and Requirements Spaces 85
viii
Chapter 4. DUALITY THEORY 89
4.1 The Symmetric Dual 89
4.2 Unsymmetric Duals 92
4.3 Duality Theorems 95
Chapter 5. THE THEORY OF LINEAR PROGRAMMING 111
5.1 Finding Primal Basic Feasible Solutions 111
5.2 The Reduced Primal Problem 114
5.3 The Primal Optimality Criterion 115
5.4 Constructing the Dual Solution 116
5.5 The Primal Simplex Method 125
5.6 Degenerate Basic Feasible Solutions 131
5.7 Unbounded Solutions Reexamined 135
5.8 Multiple Optimal Solutions 137
Chapter 6. DUALITY THEORY REVISITED 141
6.1 The Geometry of Duality and Optimality 141
6.2 Lagrangian Saddle Points and Primal Optimality 160
Chapter 7. COMPUTATIONAL ASPECTS OF LINEAR PROGRAMMING 169
7.1 The Primal Simplex Method Reexamined 169
7.2 Improving a Basic Feasible Solution 174
7.3 The Cases of Multiple Optimal, Unbounded, and
Degenerate Solutions 180
7.4 Summary of the Primal Simplex Method 187
7.5 Obtaining the Optimal Dual Solution From the Optimal
Primal Matrix 189
Chapter 8. ONE-PHASE, TWO-PHASE, AND COMPOSITE METHODS OF
LINEAR PROGRAMMING 195
8.1 Artificial Variables 195
8.2 The One-Phase Method 198
8.3 Inconsistency and Redundancy 203
8.4 Unbounded Solutions to the Artificial Problem 211
8.5 The Two-Phase Method 213
8.6 Obtaining the Optimal Primal Solution from the Optimal Dual
Matrix 221
8.7 The Composite Simplex Method 225
IX
Chapter 9. COMPUTATIONAL ASPECTS OF LINEAR PROGRAMMING:
SELECTED TRANSFORMATIONS 233
9.1 Minimizing the Objective Function 233
9.2 Unrestricted Variables 234
9.3 Bounded Variables 236
9.4 Interval Linear Programming 247
9.5 Absolute Value Functionals 249
Chapter 10. THE DUAL SIMPLEX, PRIMAL-DUAL, AND
COMPLEMENTARY PIVOT METHODS 251
10.1 Dual Simplex Method 252
10.2 Computational Aspects of the Dual Simplex Method 256
10.3 Dual Degeneracy 259
10.4 Summary of the Dual Simplex Method 259
10.5 Generating an Initial Primal-Optimal Basic Solution:
The Artificial Constraint Method 261
10.6 Primal-Dual Method 264
10.7 Summary of the Primal-Dual Method 274
10.8 A Robust Primal-Dual Algorithm 276
10.9 The Complementary Pivot Method 278
Chapter 11. POSTOPTIMALITY ANALYSIS I 289
11.1 Sensitivity Analysis 289
11.2 Structural Changes 314
Chapter 12. POSTOPTIMALITY ANALYSIS II 319
12.1 Parametric Analysis 319
12.2 The Primal-Dual Method Revisited 334
Chapter 13. INTERIOR POINT METHODS 341
13.1 Optimization Over a Sphere 341
13.2 An Overview of Karmarkar's Algorithm 343
13.3 The Projective Transformation T(X) 345
13.4 The Transformed Problem 348
13.5 Potential Function Improvement and Computational
Complexity 351
13.6 A Summary of Karmarkar's Algorithm 354
13.7 Transforming a General Linear Program to
Karmarkar Standard Form 355
x
13.8 Extensions and Modifications of Karmarkar's Algorithm 359
13.9 Methods Related to Karmarkar's Routine: Affine Scaling
Scaling Algorithms 368
13.10 Methods Related to Karmarkar's Routine: A Path-Following
Following Algorithm 379
13.11 Methods Related to Karmarkar's Routine: Potential
Reduction Algorithms 398
13.12 Methods Related to Karmarkar's Routine: A Homogeneous
and Self-Dual Interior-Point. Method 424
Chapter 14. INTERIOR POINT ALGORITHMS FOR SOLVING
LINEAR COMPLEMENTARITY PROBLEMS 433
14.1 Introduction 433
14.2 An Interior-Point, Path-Following Algorithm for
LCP ( q, M) 436
14.3 An Interior-Point, Potential-Reduction Algorithm for
LCP ( q, M) 439
14.4 A Predictor-Corrector Algorithm for Solving
LCP ( q, M) 445
14.5 Large-Step Interior-Point Algorithms for Solving
LCP(q, M) 451
14.6 Related Methods for Solving LCP { q, M) 454
Appendix A: Updating the Basis Inverse 459
Appendix B: Steepest Edge Simplex Mcthods 461
Appendix C: Derivation of the Projection Matrix 467
Refercnces 473
Notation Index 485
Index 489
