Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

Etukansi
Springer Science & Business Media, 2000 - 202 sivua
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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Sisältö

Totally Convex Functions
1
12 The Modulus of Total Convexity
17
13 Total Versus Locally Uniform Convexity
30
14 Particular Totally Convex Functions
45
Computation of Fixed Points
65
22 Totally Nonexpansive Families of Operators
79
23 Stochastic Convex Feasibility Problems
92
24 Applications in Particular Banach Spaces
109
Infinite Dimensional Optimization
129
32 Convergence of the Proximal Point Method
138
33 The Basics of a Duality Theory
145
34 An Augmented Lagrangian Method
154
35 Unconstrained Convex Minimization
171
Bibliography
189
Index
201
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Suositut otteet

Sivu 197 - Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J.
Sivu 198 - Kartsatos, Athanassios G. (ed.), Theory and applications of nonlinear operators of accretive and monotone type.
Sivu 194 - K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York 1984.
Sivu 191 - A generalized proximal point algorithm for the variational inequality problem in a Hilbert space.
Sivu 191 - Browder, FE and Petryshyn, WV The solution by iteration of nonlinear functional equations in Banach spaces, Bull.
Sivu 193 - Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,
Sivu 190 - HH Bauschke and JM Borwein. Legendre functions and the method of random Bregman projections.
Sivu 190 - Bregman, LM The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7 (1967) 200217.
Sivu 193 - A Parallel Projection Method for Finding a Common Point of a Family of Convex Sets,

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