Self Regularity

by Peng, Jiming; Roos, Cornelis; Terlaky, Tamas

• ISBN13: 9780691091938
• ISBN10: 0691091935
• Format: Paperback
• Copyright: 2002-10-07
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Preface	vii
Acknowledgements	ix
Notation	xi
List of Abbreviations	xv
Introduction and Preliminaries	1	(26)
Historical Background of Interior-Point Methods	2	(3)
Prelude	2	(1)
A Brief Review of Modern Interior-Point Methods	3	(2)
Primal-Dual Path-Following Algorithm for LO	5	(11)
Primal-Dual Model for LO, Duality Theory and the Central Path	5	(3)
Primal-Dual Newton Method for LO	8	(4)
Strategies in Path-following Algorithms and Motivation	12	(4)
Preliminaries and Scope of the Monograph	16	(11)
Preliminary Technical Results	16	(4)
Relation Between Proximities and Search Directions	20	(2)
Contents and Notational Abbreviations	22	(5)
Self-Regular Functions and Their Properties	27	(20)
An Introduction to Univariate Self-Regular Functions	28	(7)
Basic Properties of Univariate Self-Regular Functions	35	(7)
Relations Between S-R and S-C Functions	42	(5)
Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities	47	(20)
Self-Regular Functions in Rn++ and Self-Regular Proximities for LO	48	(4)
The Algorithm	52	(3)
Estimate of the Proximity After a Newton Step	55	(6)
Complexity of the Algorithm	61	(2)
Relaxing the Requirement on the Proximity Function	63	(4)
Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximities	67	(32)
Introduction to CPs and the Central Path	68	(4)
Preliminary Results on P*(k) Mappings	72	(8)
New Search Directions for P*(k) CPs	80	(3)
Complexity of the Algorithm	83	(16)
Ingredients for Estimating the Proximity	83	(4)
Estimate of the Proximity After a Step	87	(9)
Complexity of the Algorithm for CPs	96	(3)
Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities	99	(26)
Introduction to SDO, Duality Theory and Central Path	100	(3)
Preliminary Results on Matrix Functions	103	(8)
New Search Directions for SDO	111	(6)
Scaling Schemes for SDO	111	(1)
Intermezzo: A Variational Principle for Scaling	112	(2)
New Proximities and Search Directions for SDO	114	(3)
New Polynomial Primal-Dual IPMs for SDO	117	(8)
The Algorithm	117	(1)
Complexity of the Algorithm	118	(7)
Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities	125	(34)
Introduction to SOCO, Duality Theory and The Central Path	126	(3)
Preliminary Results on Functions Associated with Second-Order Cones	129	(13)
Jordan Algebra, Trace and Determinant	130	(2)
Functions and Derivatives Associated with Second-Order Cones	132	(10)
New Search Directions for SOCO	142	(8)
Preliminaries	142	(1)
Scaling Schemes for SOCO	143	(2)
Intermezzo: A Variational Principle for Scaling	145	(2)
New Proximities and Search Directions for SOCO	147	(3)
New IPMs for SOCO	150	(9)
The Algorithm	150	(2)
Complexity of the Algorithm	152	(7)
Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimization	159	(10)
The Self-Dual Embedding Model for LO	160	(2)
The Embedding Model for CP	162	(3)
Self-Dual Embedding Models for SDO and SOCO	165	(4)
Conclusions	169	(6)
A Survey of the Results and Future Research Topics	170	(5)
References	175	(8)
Index	183