Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization
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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Totally Convex Functions
12 The Modulus of Total Convexity
13 Total Versus Locally Uniform Convexity
14 Particular Totally Convex Functions
Computation of Fixed Points
22 Totally Nonexpansive Families of Operators
23 Stochastic Convex Feasibility Problems
24 Applications in Particular Banach Spaces
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Totally Convex Functions for Fixed Points Computation and Infinite ...
D. Butnariu,A.N. Iusem
Rajoitettu esikatselu - 2012
according to Proposition algorithms applied augmented Lagrangian method Banach space bounded sets Bregman distance Bregman function computational continuously differentiable contradiction converges to zero converges weakly convex feasibility problems convex optimization convex set deduce defined Denote Df(z exists a positive f is continuous f is totally family of operators finite function f Gâteaux derivative Hence Hilbert space holds implies infimum infinite dimensional Int(D Int(Dom(f Int(TP k-oo last inequality Lebesgue measure Lemma liminf locally uniformly convex lower semicontinuous modulus of total negentropy nonempty nonexpansive with respect nonexpansivity pole nonnegative Observe optimal solution orbit positive real number primal optimization problem problem 3.9 prove proximal point method respect to f sequence x'}keN sequentially consistent sets Q stochastic convex feasibility strictly convex subgradient method subsets Suppose totally convex functions totally nonexpansive operators weak accumulation points weak convergence
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