LARGE SCALE LINEAR AND INTEGER OPTIMIZATION: A UNIFIED APPROACH

Transcription

1 LARGE SCALE LINEAR AND INTEGER OPTIMIZATION: A UNIFIED APPROACH Richard Kipp Martin Graduate School of Business University of Chicago % Kluwer Academic Publishers Boston/Dordrecht/London

2 CONTENTS Preface Part I MOTIVATION l 1 LINEAR AND INTEGER LINEAR OPTIMIZATION Introduction Linear and Integer Linear Optimization A Guided Tour of Applications Special Structure Linear and Integer Linear Programming Codes Other Directions Exercises 29 Part II THEORY 33 2 LINEAR SYSTEMS AND PROJECTION Introduction Projection for Equality Systems: Gaussian Elimination Projection for Inequality Systems: Fourier-Motzkin Elimination Applications of Projection Theorems of the Alternative Duality Theory Complementary Slackness Sensitivity Analysis 65 * 2.9 Conclusion Exercises 75 xv

3 viii LARGE SCALE LINEAR AND INTEGER OPTIMIZATION 3 LINEAR SYSTEMS AND INVERSE PROJECTION Introduction Deleting Constraints by Adding Variables Dual Relationships Sensitivity Analysis Conclusion Homework Exercises INTEGER LINEAR SYSTEMS: PROJECTION AND INVERSE PROJECTION Introduction Background Material Solving A System of Congruence Equations Integer Linear Equalities Integer Linear Inequalities: Projection Integer Linear Inequalities: Inverse Projection Conclusion Exercises 137 Part III ALGORITHMS THE SIMPLEX ALGORITHM Introduction Motivation Pivoting Revised Simplex Product Form of the Inverse Degeneracy and Cycling Complexity of the Simplex Algorithm Conclusion Exercises MORE ON SIMPLEX Introduction Sensitivity Analysis The Dual Simplex Algorithm 191

4 Contents ix 6.4 Simple Upper Bounds and Special Structure Finding a Starting Basis Pivot Column Selection Other Computational Issues Conclusion Exercises INTERIOR POINT ALGORITHMS: POLYHEDRAL TRANSFORMATIONS Introduction Projective Transformations Karmarkar's Algorithm Polynomial Termination Purification, Standard Form and Sliding Objective Affine Polyhedral Transformations Geometry of the Least Squares Problem Conclusion Exercises INTERIOR POINT ALGORITHMS: BARRIER METHODS Introduction Primal Path Following Dual Path Following Primal-Dual Path Following Polynomial Termination of Path Following Algorithms Relation to Polyhedral Transformation Algorithms Predictor-Corrector Algorithms Other Issues Conclusion Exercises INTEGER PROGRAMMING Introduction Modeling with Integer Variables Branch-and-Bound Node and Variable Selection 324

5 x LARGE SCALE LINEAR AND INTEGER OPTIMIZATION 9.5 More General Branching Conclusion Exercises 341 Part IV SOLVING LARGE SCALE PROBLEMS: DECOMPOSITION METHODS PROJECTION: BENDERS' DECOMPOSITION Introduction The Benders' Algorithm A Location Application Dual Variable Selection Conclusion Exercises INVERSE PROJECTION: DANTZIG-WOLFE DECOMPOSITION Introduction Dantzig-Wolfe Decomposition A Location Application Taking Advantage of Block Angular Structure Computational Issues Conclusion Exercises LAGRANGIAN METHODS Introduction The Lagrangian Dual Extension to Integer Programming Properties of the Lagrangian Dual Optimizing the Lagrangian Dual Computational Issues A Decomposition Algorithm for Integer Programming Conclusion Exercises 435

6 Contents xi Part V SOLVING LARGE SCALE PROBLEMS: USING SPECIAL STRUCTURE SPARSE METHODS Introduction LU Decomposition Sparse LU Update Numeric Cholesky Factorization Symbolic Cholesky Factorization Storing Sparse Matrices Programming Issues Computational Results: Barrier versus Simplex Conclusion Exercises NETWORK FLOW LINEAR PROGRAMS Introduction Totally Unimodular Linear Programs Network Simplex Algorithm Important Network Flow Problems Almost Network Problems Integer Polyhedra Conclusion Exercises LARGE INTEGER PROGRAMS: PREPROCESSING AND CUTTING PLANES Formulation Principles and Techniques Preprocessing Cutting Planes Branch-and-Cut Lifting Lagrangian Cuts Integer Programming Test Problems Conclusion Exercises 563

7 xii LARGE SCALE LINEAR AND INTEGER OPTIMIZATION 16 LARGE INTEGER PROGRAMS: PROJECTION AND INVERSE PROJECTION Introduction Auxiliary Variable Methods A Projection Theorem Branch-and-Price Projection of Extended Formulations: Benders' Decomposition Revisited Conclusion Exercises 630 Part VI APPENDIX 633 A POLYHEDRAL THEORY 635 А.1 Introduction 635 A.2 Concepts and Definitions 635 А.З Faces of Polyhedra 640 A.4 Finite Basis Theorems 645 A.5 Inner Products, Subspaces and Orthogonal Subspaces 651 A.6 Exercises 653 В COMPLEXITY THEORY 657 B.l Introduction 657 B.2 Solution Sizes 660 B.3 The Turing Machine 661 B.4 Complexity Classes 663 B.5 Satisfiability 667 B.6 ЛЛР-Completeness 669 B.7 Complexity of Gaussian Elimination 670 B.8 Exercises 674 С BASIC GRAPH THEORY 677 D SOFTWARE AND TEST PROBLEMS 681 E NOTATION 683

8 Contents xüi References 687 AUTHOR INDEX 723 TOPIC INDEX 731