Self-Regularity

A New Paradigm for Primal-Dual Interior-Point Algorithms

Jiming Peng, Cornelis Roos, Tamás Terlaky

198 pages, parution le 16/12/2002

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Résumé

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.

Contents

Chapter 1. Introduction and Preliminaries 1
Chapter 2. Self-Regular Functions and Their Properties
Chapter 3. Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities
Chapter 4. Interior-Point Methods for Complementarity Problems Based on Self-Regular
Chapter 5. Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities
Chapter 6. Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities 125
Chapter 7. Initialization: Embedding Models for Linear Optimization, Complementarity Problems,
Chapter 8. Conclusions

L'auteur - Jiming Peng

Jiming Peng is Professor of Mathematics at McMaster University and has published widely on nonlinear programming and interior-points methods.

L'auteur - Cornelis Roos

Cornelis Roos holds joint professorships at Delft University of Technology and Leiden University. He is an editor of several journals, coauthor of more than 100 papers, and coauthor (with Tamás Terlaky and Jean-Philippe Vial) of Theory and Algorithms for Linear Optimization.

L'auteur - Tamás Terlaky

Tamás Terlaky is Professor in the Department of Computing and Software at McMaster University, founding Editor in Chief of Optimization and Engineering, coauthor of more than 100 papers, and an editor of several journals and two books.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Jiming Peng, Cornelis Roos, Tamás Terlaky
Parution	16/12/2002
Nb. de pages	198
Format	15,5 x 23,5
Couverture	Broché
Poids	305g
Intérieur	Noir et Blanc
EAN13	9780691091938
ISBN13	978-0-691-09193-8