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

By Jiming Peng; Cornelis Roos et al. | Go to book overview

Preface

The primary goal of this monograph is to introduce a new framework for the theory of primal-dual interior-point methods (IPMs) based on the notion of the self-regularity of a function. Starting from Karmarkar's epoch-making paper [52] in 1984, the research on IPMs has dominated the field of continuous optimization for more than 15 years, and over 3000 papers have been published relating to IPMs. With the efforts of many excellent experts in the field, particularly the path-breaking work by Nesterov and Nemirovskii [83], the theory of IPMs has been developed into a rather mature principle.

An important concept in the IPM literature is the central path of the underlying problem. This was first recognized by Sonnevend [102] and Megiddo [70]. The primal-dual framework presented in [70] laid down the bedrock of many practically efficient IPMs with interesting theoretical properties. These IPMs utilize various strategies to follow the central path approximately. Two contrasting approaches in the analysis and implementation of IPMs are the socalled small-update (with small neighborhood) and large-update (with large neighborhood) methods. Unfortunately, there is still a big gap between the theory and the practical performance of IPMs with respect to these two strategies. As stated by Renegar [97, pp. 51], “It is one of the ironies of the IPM literature that algorithms which are more efficient in practice often have somewhat worse complexity bounds.”

The motivation for this work is to bridge this gap. We deal with linear optimization, nonlinear complementarity problems, semidefinite optimization and second-order conic optimization problems. Our framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered may be interpreted as a path-following method or a constrained potential reduction method. Starting from a primal-dual strictly feasible point, our 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 exploring extensively some intriguing properties of the self-regular function, we establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs.

-vii-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Bookmark this page
Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms
Table of contents

Table of contents

Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen
/ 185

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.