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Self-Regularity : A New Paradigm for Primal-Dual Interior-Point Algorithms - Jiming Peng


A New Paradigm for Primal-Dual Interior-Point Algorithms


Published: 7th October 2002
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Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function.

The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs.

Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.

"The new idea of self-regular functions is very elegant and I am sure that this book will have a major impact on the field of optimization."--Robert Vanderbei, Princeton University
"The progress outlined in Self-Regularity represents one of the really major events in our field during the last five years or so. This book requires just standard mathematical background on the part of the reader and is thus accessible to beginners as well as experts."--Arkadi Nemirovski, Technion-Israel Institute of Technology

Prefacep. vii
Acknowledgementsp. ix
Notationp. xi
List of Abbreviationsp. xv
Introduction and Preliminariesp. 1
Historical Background of Interior-Point Methodsp. 2
Preludep. 2
A Brief Review of Modern Interior-Point Methodsp. 3
Primal-Dual Path-Following Algorithm for LOp. 5
Primal-Dual Model for LO, Duality Theory and the Central Pathp. 5
Primal-Dual Newton Method for LOp. 8
Strategies in Path-following Algorithms and Motivationp. 12
Preliminaries and Scope of the Monographp. 16
Preliminary Technical Resultsp. 16
Relation Between Proximities and Search Directionsp. 20
Contents and Notational Abbreviationsp. 22
Self-Regular Functions and Their Propertiesp. 27
An Introduction to Univariate Self-Regular Functionsp. 28
Basic Properties of Univariate Self-Regular Functionsp. 35
Relations Between S-R and S-C Functionsp. 42
Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximitiesp. 47
Self-Regular Functions inR n + + and Self-Regular Proximities for LOp. 48
The Algorithmp. 52
Estimate of the Proximity After a Newton Stepp. 55
Complexity of the Algorithmp. 61
Relaxing the Requirement on the Proximity Functionp. 63
Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximitiesp. 67
Introduction to CPs and the Central Pathp. 68
Preliminary Results on P * (k) Mappingsp. 72
New Search Directions for P * (k) CPsp. 80
Complexity of the Algorithmp. 83
Ingredients for Estimating the Proximityp. 83
Estimate of the Proximity After a Stepp. 87
Complexity of the Algorithm for CPsp. 96
Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximitiesp. 99
Introduction to SDO, Duality Theory and Central Pathp. 100
Preliminary Results on Matrix Functionsp. 103
New Search Directions for SDOp. 111
Scaling Schemes for SDOp. 111
Intermezzo: A Variational Principle for Scalingp. 112
New Proximities and Search Directions for SDOp. 114
New Polynomial Primal-Dual IPMs for SDOp. 117
The Algorithmp. 117
Complexity of the Algorithmp. 118
Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximitiesp. 125
Introduction to SOCO, Duality Theory and The Central Pathp. 126
Preliminary Results on Functions Associated with Second-Order Conesp. 129
Jordan Algebra, Trace and Determinantp. 130
Functions and Derivatives Associated with Second-Order Conesp. 132
New Search Directions for SOCOp. 142
Preliminariesp. 142
Scaling Schemes for SOCOp. 143
Intermezzo: A Variational Principle for Scalingp. 145
New Proximities and Search Directions for SOCOp. 147
New IPMs for SOCOp. 150
The Algorithmp. 150
Complexity of the Algorithmp. 152
Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimizationp. 159
The Self-Dual Embedding Model for LOp. 160
The Embedding Model for CPp. 162
Self-Dual Embedding Models for SDO and SOCOp. 165
Conclusionsp. 169
A Survey of the Results and Future Research Topicsp. 170
Referencesp. 175
Indexp. 183
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691091938
ISBN-10: 0691091935
Series: Princeton Series in Applied Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 208
Published: 7th October 2002
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.27
Weight (kg): 0.03