Arc-Search Techniques for Interior-Point Methods

The interior-point method is a numerical optimization technique that has become popular since the 1980s, but has been plagued by a gap between theoretical polynomial bounds and efficiency. This book discusses recent developments that resolve this dilemma, presenting three major parts: an analysis of important algorithms, arc-search techniques, and extensions to more general problems.

Format: Paperback / softback
Length: 306 pages
Publication date: 01 August 2022
Publisher: Taylor & Francis Ltd

This book delves into a crucial aspect of numerical optimization known as the interior-point method, which has gained significant attention since the 1980s. Initially, it was recognized that all simplex algorithms were not convergent in polynomial time, leading to the exploration of interior-point algorithms that could guarantee convergence in polynomial time. However, a significant gap persisted between the theoretical polynomial bounds of these algorithms and their actual computational efficiency. Strategies that were crucial for enhancing computational speed became barriers in establishing strong polynomial bounds. The more these strategies were employed in algorithms, the worse the polynomial bounds deteriorated. To exacerbate the problem, Mehrotra's predictor-corrector (MPC) algorithm, which was once the most popular and efficient interior-point algorithm, failed to prove convergence despite utilizing all known good strategies. This lack of polynomiality, a fundamental requirement of the simplex method, posed a significant challenge.

In response to this dilemma, this book presents recent developments that offer a solution. It is organized into three major parts. The first part, comprising Chapters 1, 2, 3, and 4, explores some of the most important algorithms developed during the 1990s, many of which are widely recognized. The primary objective of this section is to analyze the polynomial bounds of these algorithms and summarize the computational experiences associated with them. By examining these algorithms, the book sheds light on the dilemma described earlier and highlights the challenges encountered in achieving good polynomial bounds.

The second part, consisting of Chapters 5, 6, 7, and 8, takes a step-by-step approach to resolving the dilemma. It introduces arc-search techniques, which are powerful tools for finding optimal solutions to optimization problems. Through these techniques, the book demonstrates how to overcome the barriers encountered in proving strong polynomial bounds and presents an efficient algorithm with the lowest polynomial bound.

The third and final part, comprising Chapters 9, 10, 11, and 12, extends the arc-search techniques to address more general problems. It explores various extensions and adaptations of arc-search algorithms, demonstrating their versatility and applicability in various optimization scenarios.

In conclusion, this book provides a comprehensive and up-to-date exploration of the interior-point method, addressing the dilemma of achieving good polynomial bounds while maintaining computational efficiency. By presenting recent developments and employing arc-search techniques, it offers a solution that bridges the gap between theory and practice, paving the way.

Weight: 580g
Dimension: 234 x 156 (mm)
ISBN-13: 9780367510091