Mathematical View of Interior-Point Methods in Convex Optimization

ISBN-10: 0898715024

ISBN-13: 9780898715026

Edition: 2001

Authors: James Renegar

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Description:

This compact book, through the simplifying perspective it presents, will take a reader who knows little of interior-point methods to within sight of the research frontier, developing key ideas that were over a decade in the making by numerous interior-point method researchers. It aims at developing a thorough understanding of the most general theory for interior-point methods, a class of algorithms for convex optimization problems. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years. In that time, the theory has matured tremendously, but much of the literature is difficult to understand, even for specialists. By focusing only on essential elements of the theory and emphasizing the underlying geometry, A Mathematical View of Interior-Point Methods in Convex Optimization makes the theory accessible to a wide audience, allowing them to quickly develop a fundamental understanding of the material.
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Book details

List price: $39.50
Copyright year: 2001
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 125
Size: 6.75" wide x 9.75" long x 0.50" tall
Weight: 0.550
Language: English

Preface
Preliminaries
Linear Algebra
Gradients
Hessians
Convexity
Fundamental Theorems of Calculus
Newton's Method
Basic Interior-Point Method Theory
Intrinsic Inner Products
Self-Concordant Functionals
Introduction
Self-Concordancy and Newton's Method
Other Properties
Barrier Functionals
Introduction
Analytic Centers
Optimal Barrier Functionals
Other Properties
Logarithmic Homogeneity
Primal Algorithms
Introduction
The Barrier Method
The Long-Step Barrier Method
A Predictor-Corrector Method
Matters of Definition
Conic Programming and Duality
Conic Programming
Classical Duality Theory
The Conjugate Functional
Duality of the Central Paths
Self-Scaled (or Symmetric) Cones
Introduction
An Important Remark on Notation
Scaling Points
Gradients and Norms
A Useful Theorem
The Nesterov--Todd Directions
Primal-Dual Path-Following Methods
Measures of Proximity
An Algorithm
Another Algorithm
A Primal-Dual Potential-Reduction Method
The Potential Function
The Algorithm
The Analysis
Bibliography
Index
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