Nonlinear Conjugate Gradient Methods for Unconstrained Optimization

The book discusses two approaches for solving large-scale unconstrained optimization problems: the limited-memory quasi-Newton method and the conjugate gradient method. It provides a detailed explanation of conjugate gradient methods, including linear, standard, and hybrid methods, as well as memoryless BFGS preconditioned methods. The convergence analysis, computational performances, and comparisons to other methods are discussed for each approach. Numerous numerical studies are provided to demonstrate the effectiveness of conjugate gradient algorithms in solving a variety of optimization problems. The book is intended for mathematical programming researchers, theorists, practitioners in operations research, engineering, and industry, as well as graduate students in mathematics and mathematical programming.

Format: Paperback / softback
Length: 498 pages
Publication date: 24 June 2021
Publisher: Springer Nature Switzerland AG

Two well-known methods for tackling large-scale, unconstrained optimization problems are the limited-memory quasi-Newton method (truncated Newton method) and the conjugate gradient method. This groundbreaking book is the first to delve into conjugate gradient methods, providing a comprehensive exploration of their properties, convergence characteristics, and remarkable performance in solving complex, large-scale unconstrained optimization problems across various applications. In addition to comparing these methods to the limited-memory and truncated Newton methods, the book delves into various topics in depth, including linear conjugate gradient methods, standard conjugate gradient methods, acceleration techniques, hybrid approaches, modifications of the standard scheme, memoryless BFGS preconditioning, and three-term methods. Furthermore, the book addresses other conjugate gradient methods that employ clustering of eigenvalues or the minimization of the condition number of the iteration matrix. For each method, thorough convergence analysis, computational performance evaluations, and comparisons to other conjugate gradient methods are presented. The theory behind the conjugate gradient algorithms is developed with a clear, rigorous, and user-friendly exposition, empowering readers to gain a deep understanding of their properties, convergence, and the ability to develop and prove their convergence for their own methods. Numerous numerical studies are included, showcasing the behavior of conjugate gradient algorithms in solving a diverse collection of 800 unconstrained optimization problems with varying structures and complexities, involving variables ranging from 1000 to 10000. This book is a valuable resource for researchers, practitioners, and students interested in advancing their knowledge and expertise in developing and applying innovative techniques for solving complex optimization problems. It serves as a comprehensive guide for those seeking to leverage the power of conjugate gradient methods in their work.

Weight: 801g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030429522
Edition number: 1st ed. 2020