Newton-type methods for optimization and variational problems.

*(English)*Zbl 1304.49001
Springer Series in Operations Research and Financial Engineering. Cham: Springer (ISBN 978-3-319-04246-6/hbk; 978-3-319-04247-3/ebook). xix, 573 p. (2014).

This book is devoted to the theoretical analysis of the fundamental Newtonian and Newtonian-related approaches in order to solve optimization and variational problems. A central focus is the relationship between the basic Newton scheme for a given problem and algorithms that also enjoy fast local convergence. The authors develop general perturbed Newtonian frameworks that preserve fast convergence and consider specific algorithms as particular cases within those frameworks.

Some material presented in this book is a product of the authors’ research over the past 10 years or so. The book consists of 7 Chapters.

The first Chapter collects a summary of basic facts of optimization theory and optimality conditions. It introduces various classes of variational problems and discusses their stability, sensitivity properties, regularity conditions and error bounds. Elements of nonsmooth analysis are presented as well.

Chapter 2 contains an introduction to Newton methods on systems of nonlinear equations and unconstrained optimization. This chapter contains also some materials on line search and trust-region globalisation techniques, quasi-Newton methods and semismooth Newton methods.

Chapter 3 is devoted to local convergence analysis of methods for variational problems. First the fundamental Josephy-Newton method for generalized equations is considered. This chapter also includes the semismooth Newton methods for equations arising from reformulations of complementarity conditions and active-set Newton methods.

Chapter 4 deals with local analysis of constrained optimization algorithms. Convergence of the fundamental Sequential Quadratic Programming (SQP) method is derived by relating it to the Josephy-Newton framework for generalized equations. Then, the perturbed SQP framework is introduced which allows to treat in a unified manner various modifications of SQP and some methods that are not in the SQP class.

Chapter 5 provides line search globalization of some previously discussed local methods for variational problems.

Chapter 6 discusses line search globalization of SQP based on merit functions, including the Maratos effect and two tools for preserving fast local convergence rate (using the nonsmooth augmented Lagrangian as the merit function and using second order corrections for the step). Finally, an alternative generic filter globalization for constrained optimization and a trust region filter SQP method are presented.

Chapter 7 deals with Newton-type methods for degenerate problems i.e. problems with nonisolated solutions.

I think that this volume will be very useful for graduate students and researchers in the field of optimization and variational problems as it provides a unified view of classical as well as very recent developments in the field of Newton-type methods for optimization and variational problems.

Some material presented in this book is a product of the authors’ research over the past 10 years or so. The book consists of 7 Chapters.

The first Chapter collects a summary of basic facts of optimization theory and optimality conditions. It introduces various classes of variational problems and discusses their stability, sensitivity properties, regularity conditions and error bounds. Elements of nonsmooth analysis are presented as well.

Chapter 2 contains an introduction to Newton methods on systems of nonlinear equations and unconstrained optimization. This chapter contains also some materials on line search and trust-region globalisation techniques, quasi-Newton methods and semismooth Newton methods.

Chapter 3 is devoted to local convergence analysis of methods for variational problems. First the fundamental Josephy-Newton method for generalized equations is considered. This chapter also includes the semismooth Newton methods for equations arising from reformulations of complementarity conditions and active-set Newton methods.

Chapter 4 deals with local analysis of constrained optimization algorithms. Convergence of the fundamental Sequential Quadratic Programming (SQP) method is derived by relating it to the Josephy-Newton framework for generalized equations. Then, the perturbed SQP framework is introduced which allows to treat in a unified manner various modifications of SQP and some methods that are not in the SQP class.

Chapter 5 provides line search globalization of some previously discussed local methods for variational problems.

Chapter 6 discusses line search globalization of SQP based on merit functions, including the Maratos effect and two tools for preserving fast local convergence rate (using the nonsmooth augmented Lagrangian as the merit function and using second order corrections for the step). Finally, an alternative generic filter globalization for constrained optimization and a trust region filter SQP method are presented.

Chapter 7 deals with Newton-type methods for degenerate problems i.e. problems with nonisolated solutions.

I think that this volume will be very useful for graduate students and researchers in the field of optimization and variational problems as it provides a unified view of classical as well as very recent developments in the field of Newton-type methods for optimization and variational problems.

Reviewer: Basilis Kokkinis (Athens)