Self-regularity : a new paradigm for primal-dual interior-point algorithms /
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| Author / Creator: | Peng, Jiming. |
|---|---|
| Imprint: | Princeton, N.J. : Chichester : Princeton University Press, 2002. |
| Description: | 176 p. ; 23 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4775919 |
Table of Contents:
- Preface
- Acknowledgements
- Notation
- List of Abbreviations
- Chapter 1. Introduction and Preliminaries
- 1.1. Historical Background of Interior-Point Methods
- 1.1.1. Prelude
- 1.1.2. A Brief Review of Modern Interior-Point Methods
- 1.2. Primal-Dual Path-Following Algorithm for LO
- 1.2.1. Primal-Dual Model for LO, Duality Theory and the Central Path
- 1.2.2. Primal-Dual Newton Method for LO
- 1.2.3. Strategies in Path-following Algorithms and Motivation
- 1.3. Preliminaries and Scope of the Monograph
- 1.3.1. Preliminary Technical Results
- 1.3.2. Relation Between Proximities and Search Directions
- 1.3.3. Contents and Notational Abbreviations
- Chapter 2. Self-Regular Functions and Their Properties
- 2.1. An Introduction to Univariate Self-Regular Functions
- 2.2. Basic Properties of Univariate Self-Regular Functions
- 2.3. Relations Between S-R and S-C Functions
- Chapter 3. Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities
- 3.1. Self-Regular Functions inR n + + and Self-Regular Proximities for LO
- 3.2. The Algorithm
- 3.3. Estimate of the Proximity After a Newton Step
- 3.4. Complexity of the Algorithm
- 3.5. Relaxing the Requirement on the Proximity Function
- Chapter 4. Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximities
- 4.1. Introduction to CPs and the Central Path
- 4.2. Preliminary Results on P * (k) Mappings
- 4.3. New Search Directions for P * (k) CPs
- 4.4. Complexity of the Algorithm
- 4.4.1. Ingredients for Estimating the Proximity
- 4.4.2. Estimate of the Proximity After a Step
- 4.4.3. Complexity of the Algorithm for CPs
- Chapter 5. Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities
- 5.1. Introduction to SDO, Duality Theory and Central Path
- 5.2. Preliminary Results on Matrix Functions
- 5.3. New Search Directions for SDO
- 5.3.1. Scaling Schemes for SDO
- 5.3.2. Intermezzo: A Variational Principle for Scaling
- 5.3.3. New Proximities and Search Directions for SDO
- 5.4. New Polynomial Primal-Dual IPMs for SDO
- 5.4.1. The Algorithm
- 5.4.2. Complexity of the Algorithm
- Chapter 6. Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities
- 6.1. Introduction to SOCO, Duality Theory and The Central Path
- 6.2. Preliminary Results on Functions Associated with Second-Order Cones
- 6.2.1. Jordan Algebra, Trace and Determinant
- 6.2.2. Functions and Derivatives Associated with Second-Order Cones
- 6.3. New Search Directions for SOCO
- 6.3.1. Preliminaries
- 6.3.2. Scaling Schemes for SOCO
- 6.3.3. Intermezzo: A Variational Principle for Scaling
- 6.3.4. New Proximities and Search Directions for SOCO
- 6.4. New IPMs for SOCO
- 6.4.1. The Algorithm
- 6.4.2. Complexity of the Algorithm
- Chapter 7. Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimization
- The Self-Dual Embedding Model for LO
- The Embedding Model for CP
- Self-Dual Embedding Models for SDO and SOCO
- Chapter 8. Conclusions
- 8.1. A Survey of the Results and Future Research Topics
- References
- Index
