Linear Programming: Mathematics, Theory and Algorithms

Transcription

1 Linear Programming: Mathematics, Theory and Algorithms

2 Applied Optimization Volume 2 The titles published in this series are listed at the end of this volume.

3 Linear Programming: Mathematics, Theory and Algorithms by Michael 1. Panik University of Hartford KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

4 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: : / e-isbn-13: Linear programming. I. Title. II. Series. T57.74.P dc 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 1 st 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.

5 In Memory of Alice E. Bourneuf and Ann F. Friedlaender

6 TABLE OF CONTENTS Chapter 1. INTRODUCTION AND OVERVIEW 1 Chapter 2. PRELIMINARY MATHEMATICS Vectors in R n Rank and Linear Transformations The Solution Set of a System of Simultaneous Linear Equations Orthogonal Projections and Least Squares Solutions Point-Set Theory: Topological Properties of R n Hyperplanes and Half-Planes (-Spaces) Convex Sets Existence of Separating and Supporting Hyperplanes Convex Cones Basic Solut.ions to Lineal' Equalities Faces of Polyhedral Convex Sets: Extreme Points, Facets, and Edges Extreme Point Representation for Polyhedral Convex Sets Directions for Polyhedral Convex Sets Combined Extreme Point and Extreme Direction Representation for Polyhedral Convex Sets Resolution of Convex Polyhedra Simplexes Linear Fundionals 66 Chapter 3. INTRODUCTION TO UNEAR PROGRAMMING The Canonical Form of a Linear Programming Problem A Graphical Solution to the Lineal' Programming Problem The Standard Form of a Linear Programming Problem Properties of the Feasible Region Existence and Location of Finite Optimal Solutions Basic Feasible and Extreme Point Solutions to the Linear Programming Problem Solutions and Requirements Spaces 85

7 viii Chapter 4. DUALITY THEORY The Symmetric Dual Unsymmetric Duals Duality Theorems 95 Chapter 5. THE THEORY OF LINEAR PROGRAMMING Finding Primal Basic Feasible Solutions The Reduced Primal Problem The Primal Optimality Criterion Constructing the Dual Solution The Primal Simplex Method Degenerate Basic Feasible Solutions Unbounded Solutions Reexamined Multiple Optimal Solutions 137 Chapter 6. DUALITY THEORY REVISITED The Geometry of Duality and Optimality Lagrangian Saddle Points and Primal Optimality 160 Chapter 7. COMPUTATIONAL ASPECTS OF LINEAR PROGRAMMING The Primal Simplex Method Reexamined Improving a Basic Feasible Solution The Cases of Multiple Optimal, Unbounded, and Degenerate Solutions Summary of the Primal Simplex Method Obtaining the Optimal Dual Solution From the Optimal Primal Matrix 189 Chapter 8. ONE-PHASE, TWO-PHASE, AND COMPOSITE METHODS OF LINEAR PROGRAMMING Artificial Variables The One-Phase Method Inconsistency and Redundancy Unbounded Solutions to the Artificial Problem The Two-Phase Method Obtaining the Optimal Primal Solution from the Optimal Dual Matrix The Composite Simplex Method 225

8 IX Chapter 9. COMPUTATIONAL ASPECTS OF LINEAR PROGRAMMING: SELECTED TRANSFORMATIONS Minimizing the Objective Function Unrestricted Variables Bounded Variables Interval Linear Programming Absolute Value Functionals 249 Chapter 10. THE DUAL SIMPLEX, PRIMAL-DUAL, AND COMPLEMENTARY PIVOT METHODS Dual Simplex Method Computational Aspects of the Dual Simplex Method Dual Degeneracy Summary of the Dual Simplex Method Generating an Initial Primal-Optimal Basic Solution: The Artificial Constraint Method Primal-Dual Method Summary of the Primal-Dual Method A Robust Primal-Dual Algorithm The Complementary Pivot Method 278 Chapter 11. POSTOPTIMALITY ANALYSIS I Sensitivity Analysis Structural Changes 314 Chapter 12. POSTOPTIMALITY ANALYSIS II Parametric Analysis The Primal-Dual Method Revisited 334 Chapter 13. INTERIOR POINT METHODS Optimization Over a Sphere An Overview of Karmarkar's Algorithm The Projective Transformation T(X) The Transformed Problem Potential Function Improvement and Computational Complexity A Summary of Karmarkar's Algorithm Transforming a General Linear Program to Karmarkar Standard Form 355

9 x 13.8 Extensions and Modifications of Karmarkar's Algorithm Methods Related to Karmarkar's Routine: Affine Scaling Scaling Algorithms Methods Related to Karmarkar's Routine: A Path-Following Following Algorithm Methods Related to Karmarkar's Routine: Potential Reduction Algorithms 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 Introduction An Interior-Point, Path-Following Algorithm for LC P( q, M) An Interior-Point, Potential-Reduction Algorithm for LC P( q, M) A Predictor-Corrector Algorithm for Solving LC P( q, M) Large-Step Interior-Point Algorithms for Solving LCP(q, M) Related Methods for Solving LC P{ 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

10 ACKNOWLEDGEMENT Anyone familiar with the area of linear programming cannot help but marvel at its richness, elegance, and diversity. These special qualities of linear programming are by no means accidental. Many extremely talented and dedicated individuals helped cultivate these attributes. I am most grateful to those researchers who pioneered as well as contributed significantly to the development of linear programming theory and methods and whose writings have served as a valuable source of reference and inspiration. Specifically, the classic works of George Dantzig, David Gale, A. J. Goldman, Harold Kuhn, Albert Tucker, and Philip Wolfe, among others, are particularly noteworthy. More recently, my interest in the study of interior point techniques is largely due to the efforts of researchers such as Kurt Anstreicher, Robert Freund, Donald Goldfarb, Clovis Gonzaga, N. Karmarkar, Masakazu Kojima, Nimrod Megiddo, Shinji Mizuno, Michael Todd, Yinyu Ye, and Akiko Yoshise. To each of these authors, as well as to many of their contemporaries, lowe a substantial intellectual debt. A significant portion of the material presented herein stems from lecture notes developed for courses such as quantitative decision theory and advanced quantitative methods which I taught for a number of years in the A.D. Barney School of Business and Public Administration at the University of Hartford. I am grateful to the University V.B. Coffin Grant Committee and the Barney School Dean's Office for providing financial support. Finally, a special note of gratitude is extended to Marilyn Baleshiski for her accuracy, dedication, and steadfastness in typing an extremely complicated and lengthy manuscript.