Potential Function Methods for Approximately Solving Linear Programming Problems

Theory and Practice

By: Daniel Bienstock

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Hardcover | 31 August 2002

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140 Pages

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23.39 x 15.6 x 0.97

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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.

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List of Figures	p. ix
List of Tables	p. xi
Preface	p. xiii
Introduction	p. xiv
Early Algorithms	p. 1
The Flow Deviation Method	p. 3
Convergence Analysis	p. 5
Analysis of the Flow Deviation method	p. 8
Historical Perspective	p. 11
The Shahrokhi and Matula Algorithm	p. 13
The algorithm proper	p. 19
Cut metrics and minimum congestion	p. 22
Cut metrics and capacitated network design	p. 23
The Exponential Potential Function - Key Ideas	p. 27
Handling more general problems	p. 27
Handling large width	p. 28
Leveraging block-angular structure	p. 30
A basic algorithm for min-max LPs	p. 30
The first stage	p. 31
The second stage	p. 32
Computing [lambda]* to absolute tolerance	p. 33
Round-robin and randomized schemes for block-angular problems	p. 39
Basic deterministic approach	p. 41
Randomized approaches	p. 43
What is best	p. 44
Optimization and more general feasibility systems	p. 44
Width, revisited	p. 47
Alternative potential functions	p. 48
A philosophical point: why these algorithms are useful	p. 48
Recent Developments	p. 51
Oblivious rounding	p. 51
Concurrent flows	p. 54
The oracle	p. 55
The deterministic algorithm	p. 57
Handling general capacities	p. 59
Comparison with the exponential potential function method	p. 62
Lower bounds for Frank-Wolfe methods	p. 62
Lower bounds under the oracle model	p. 64
The algorithms of Garg-Konemann and Fleischer	p. 66
The Luby-Nisan algorithm and related work	p. 67
Lagrangian Relaxation, Non-Differentiable Optimization and Penalty Methods	p. 69
Bundle and cutting-plane methods	p. 70
Penalty methods	p. 70
The Volume algorithm	p. 71
Computational Experiments	p. 73
Remarks on previous work	p. 74
Outline of a generic implementation	p. 76
Basic Issues	p. 78
Choosing a block	p. 78
Choosing [tau]	p. 79
Choosing [mu]	p. 80
Choosing [alpha]	p. 81
Improving Lagrangian Relaxations	p. 83
Restricted Linear Programs	p. 86
Tightening formulations	p. 88
Computational tests	p. 89
Network Design Models	p. 90
Minimum-cost multicommodity flow problems	p. 91
Maximum concurrent flow problems	p. 93
More sophisticated network design models	p. 94
Empirical trade-off between time and accuracy	p. 97
Hitting the sweet spot	p. 98
Future work	p. 101
Frequently Asked Questions	p. 103
References	p. 107
Index	p. 111
Table of Contents provided by Ingram. All Rights Reserved.

Potential Function Methods for Approximately Solving Linear Programming Problems

Theory and Practice

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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