Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

ISBN-13: 9789401057882 / Angielski / Miękka / 2012 / 205 str.

D. Butnariu;A.N. Iusem

Opis

The main purpose of this book is to present, in a unified approach, several algorithms for fixed point computation, convex feasibility and convex optimization in infinite dimensional Banach spaces, and for problems involving, eventually, infinitely many constraints. For instance, methods like the simultaneous projection algorithm for feasibility, the proximal point algorithm and the augmented Lagrangian algorithm are rigorously formulated and analyzed in this general setting and shown to be applicable to much wider classes of problems than previously known. For this purpose, a new basic concept, total convexity', is introduced. Its properties are deeply explored, and a comprehensive theory is presented, bringing together previously unrelated ideas from Banach space geometry, finite dimensional convex optimization and functional analysis. For making our general approach possible we had to improve upon classical results like the Holder-Minkowsky inequality of "L"p. All the material is either new or very recent, and has never been organized in a book. Audience: This book will be of interest to both researchers in nonlinear analysis and to applied mathematicians dealing with numerical solution of integral equations, equilibrium problems, image reconstruction, optimal control, etc."

Spis treści

1: Totally Convex Functions.- 1.1. Convex Functions and Bregman Distances.- 1.2. The Modulus of Total Convexity.- 1.3. Total Versus Locally Uniform Convexity.- 1.4. Particular Totally Convex Functions.- 2: Computation of Fixed Points.- 2.1. Totally Nonexpansive Operators.- 2.2. Totally Nonexpansive Families of Operators.- 2.3. Stochastic Convex Feasibility Problems.- 2.4. Applications in Particular Banach Spaces.- 3: Infinite Dimensional Optimization.- 3.1. A Proximal Point Method.- 3.2. Convergence of the Proximal Point Method.- 3.3. The Basics of a Duality Theory.- 3.4. An Augmented Lagrangian Method.- 3.5. Unconstrained Convex Minimization.