عنوان

Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

(Number (ISBN

9789401057882

(Number (ISBN

9789401140669

Number

b408819

Title Proper

Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

General Material Designation

[Book]

First Statement of Responsibility

by Dan Butnariu, Alfredo N. Iusem.

Place of Publication, Distribution, etc.

Dordrecht :

Name of Publisher, Distributor, etc.

Imprint: Springer,

Date of Publication, Distribution, etc.

2000.

Series Title

Applied Optimization,

Volume Designation

40

ISSN of Series

1384-6485 ;

Text of Note

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.

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Text of Note

The aim of this work is to present in a unified approach a series of results concerning totally convex functions on Banach spaces and their applications to building iterative algorithms for computing common fixed points of mea surable families of operators and optimization methods in infinite dimen sional settings. The notion of totally convex function was first studied by Butnariu, Censor and Reich [31] in the context of the space lRR because of its usefulness for establishing convergence of a Bregman projection method for finding common points of infinite families of closed convex sets. In this finite dimensional environment total convexity hardly differs from strict convexity. In fact, a function with closed domain in a finite dimensional Banach space is totally convex if and only if it is strictly convex. The relevancy of total convexity as a strengthened form of strict convexity becomes apparent when the Banach space on which the function is defined is infinite dimensional. In this case, total convexity is a property stronger than strict convexity but weaker than locally uniform convexity (see Section 1.3 below). The study of totally convex functions in infinite dimensional Banach spaces was started in [33] where it was shown that they are useful tools for extrapolating properties commonly known to belong to operators satisfying demanding contractivity requirements to classes of operators which are not even mildly nonexpansive.

International Standard Book Number

9789401057882

Title

Springer eBooks

Discrete groups.

Functional analysis.

Integral equations.

Mathematical optimization.

Mathematics.

Operator theory.

Butnariu, Dan.

Iusem, Alfredo N.

SpringerLink (Online service)

Date of Transaction

20190307162800.0

Electronic name

[Book]

Y