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＜電子ブック＞
Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization / by D. Butnariu, A.N. Iusem
(Applied Optimization ; 40)

版	1st ed. 2000.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	2000
本文言語	英語
大きさ	XVI, 205 p : online resource
著者標目	*Butnariu, D author Iusem, A.N author SpringerLink (Online service)
件　名	LCSH:Mathematical optimization LCSH:Calculus of variations LCSH:Convex geometry  LCSH:Discrete geometry LCSH:Functional analysis LCSH:Operator theory LCSH:Integral equations FREE:Calculus of Variations and Optimization FREE:Convex and Discrete Geometry FREE:Functional Analysis FREE:Operator Theory FREE:Integral Equations
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-94-011-4066-9

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