Potential function methods for approximately solving linear programming problems

132 Pages·2002·1.98 MB·English

by  Bienstock D.

Potential Function Methods for Approximately
Solving LinearProgrammingProblems
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Bienstock, D. / Potential FunctionMethodsfor Approx. Solving LinearProgramming Problems
POTENTIAL FUNCTION METHODS FOR
APPROXIMATELY SOLVING LINEAR
PROGRAMMING PROBLEMS: THEORY
AND PRACTICE
DANIEL BIENSTOCK
Department of IEOR
Columbia University
New York
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 0-306-47626-6
Print ISBN: 1-4020-7173-6
©2002 Kluwer Academic Publishers
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Contents
List of Figures ix
List of Tables xi
Preface xiii
1 Introduction xv
1. EARLY ALGORITHMS 1
1 The Flow Deviation Method 3
1.1 Convergence Analysis 5
1.2 Analysis of the Flow Deviation method 8
1.3 Historical Perspective 11
2 The Shahrokhi and Matula Algorithm 13
2.1 The algorithm proper 19
2.2 Cut metrics and minimum congestion 22
2.2.1 Cut metrics and capacitated network design 23
2. THE EXPONENTIAL POTENTIAL
FUNCTION – KEY IDEAS 27
0.2.2 Handling more general problems 27
0.2.3 Handling large width 28
0.2.4 Leveraging block-angularstructure 30
1 A basic algorithm for min-max LPs 30
1.1 The first stage 31
1.2 The second stage 32
1.3 Computing to absolute tolerance 33
2 Round-robin and randomized schemes for block-angular
problems 39
2.1 Basic deterministic approach 41
2.2 Randomized approaches 43
vi APPROXIMATELY SOLVING LARGE LINEAR PROGRAMS
2.3 What is best 44
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
1.1 Concurrent flows 54
1.1.1 The oracle 55
1.1.2 The deterministic algorithm 57
1.1.3 Handling general capacities 59
1.1.4 Comparison with the exponential potential function
method 62
2 Lower bounds for Frank-Wolfe methods 62
2.1 Lower bounds under the oracle model 64
3 The algorithms of Garg-Könemann and Fleischer 66
3.1 The Luby-Nisan algorithm and related work 67
4 Lagrangian Relaxation, Non-Differentiable Optimization
and Penalty Methods 69
4.0.1 Bundle and cutting-plane methods 70
4.0.2 Penalty methods 70
4.0.3 The Volume algorithm 71
4. COMPUTATIONAL EXPERIMENTS 73
0.1 Remarks on previous work 74
0.2 Outline of a generic implementation 76
1 Basic Issues 78
1.1 Choosing a block 78
1.2 Choosing 79
1.3 Choosing 80
1.4 Choosing 81
2 Improving Lagrangian Relaxations 83
3 Restricted Linear Programs 86
4 Tighteningformulations 88
5 Computational tests 89
5.1 Network Design Models 90
5.2 Minimum-cost multicommodity flow problems 91
5.3 Maximum concurrent flow problems 93
Contents vii
5.4 More sophisticated network design models 94
5.5 Empirical trade-offbetween time and accuracy 97
5.6 Hitting the sweet spot 98
6 Future work 101
APPENDIX - FREQUENTLY ASKED QUESTIONS 103
References 107
Index 111
This Page Intentionally Left Blank
List of Figures
0.1 Time as a functionofcolumns fordual simplex on
concurrentflowproblems;best-fit cubic xvii
0.2 Timeasafunctionofcorrect digitsforBarriercode
oninstancenetd9 xviii
4.1 Time as a function of column count for poten-
tial function methodonRMFGEN concurrentflow
problems 95
4.2 Timeasfunctionof for instancermfgen2 99
4.3 Time as functionof for instance netd9 100