Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization
Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization by Dan Butnariu is a scholarly work published by Springer Science & Business Media in 2000. This 202-page book is written in English and delves into algorithms for fixed point computation, convex feasibility, and convex optimization within infinite dimensional Banach spaces, addressing problems that may involve infinitely many constraints.
Readers will find a unified approach to various methods, including the simultaneous projection algorithm, proximal point algorithm, and augmented Lagrangian algorithm, all rigorously formulated and analyzed. The book introduces the concept of total convexity and explores its properties, presenting a comprehensive theory that integrates ideas from Banach space geometry, finite dimensional convex optimization, and functional analysis. This edition serves as a resource for those interested in mathematical analysis, optimization, and related fields.
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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 a general approach possible the work aims to improve upon classical results like the Holder-Minkowsky inequality of ℒp.
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