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(International Series in Operations Research & Management Science): : Daniel Bienstock

Potential Function Methods For Approximately Solving Linear Programming Problems breaks new ground in linear programming theory. The book draws on the research developments in three broad areas: linear and integer programming, numerical analysis, and the computational architectures which enable speedy, high-level algorithm design. During the last ten years, a new body of research within the field of optimization research has emerged, which seeks to develop good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context of the modern theory of algorithms. The result of this work, in which Daniel Bienstock has been very much involved, has been a family of algorithms with solid theoretical foundations and with growing experimental success. This book will examine these algorithms, starting with some of the very earliest examples, and through the latest theoretical and computational developments.

Breaks ground in linear programming theory. Draws on the research developments in the areas of linear and integer programming, numerical analysis, and high-level algorithm design.

Table of Contents

List of Figures

List of Tables

Preface

1 Introduction

1 Early Algorithms 1

1 The Flow Deviation Method 3

2 The Shahrokhi and Matula Algorithm 13

2 The Exponential Potential Function - Key Ideas 27

1 A basic algorithm for min-max LPs 30

2 Round-robin and randomized schemes for block-angular problems 39

3 Optimization and more general feasibility systems 44

4 Width, revisited 47

5 Alternative potential functions 48

6 A philosophical point: why these algorithms are useful 48

3 Recent Developments 51

1 Oblivious rounding 51

2 Lower bounds for Frank-Wolfe methods 62

3 The algorithms of Garg-Konemann and Fleischer 66

4 Lagrangian Relaxation, Non-Differentiable Optimization and Penalty Methods 69

4 Computational Experiments 73

1 Basic Issues 78

2 Improving Lagrangian Relaxations 83

3 Restricted Linear Programs 86

4 Tightening formulations 88

5 Computational tests 89

6 Future work 101

App.: Frequency Asked Questions 103

References 107

Index 111

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