Optimization algorithms on matrix manifolds. (English) Zbl 1147.65043

Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008).
This book has 224 pages and it is divided into an Introduction and seven chapters, namely: Chapter 2: Motivation and applications (p. 5); Chapter 3: Matrix manifolds: First-order geometry (p. 17); Chapter 4: Line search algorithms on manifolds (p. 54); Chapter 5: Matrix manifolds: Second-order geometry (p. 91); Chapter 6: Newton’s method (p. 91); Chapter 7: Trust-region methods (p. 136); Chapter 8: A constellation of superlinear algorithms (p. 168–198).
Many problems in the science and engineering can be rephrased as optimization problems on a matrix search space endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms and to give differential geometric interpretations. Chapters 3, 4 and 5 offer differential geometry notions necessary to the algorithmic developments. The book presents applicative domains to mathematicians, engineers and computer scientists.


65K05 Numerical mathematical programming methods
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C51 Interior-point methods
90C53 Methods of quasi-Newton type
90C30 Nonlinear programming