Totally Convex Functions for Fixed Points Computation and Infinite Dimensional OptimizationSpringer Science & Business Media, 6.12.2012 - 205 sivua 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. |
Muita painoksia - Näytä kaikki
Totally Convex Functions for Fixed Points Computation and Infinite ... Dan Butnariu,Alfredo N. Iusem Rajoitettu esikatselu - 2000 |
Totally Convex Functions for Fixed Points Computation and Infinite ... D. Butnariu,A.N. Iusem Esikatselu ei käytettävissä - 2012 |
Totally Convex Functions for Fixed Points Computation and Infinite ... D. Butnariu,A.N. Iusem Esikatselu ei käytettävissä - 2011 |
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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 Dƒ(x Dƒ(y Dƒ(z dµ(w exists a positive family of operators finite dimensional function f Gâteaux derivative Hence Hilbert space holds implies Int(D ISBN last inequality Lebesgue measure Lemma lim inf lim sup locally uniformly convex lower semicontinuous modulus of total negentropy nonexpansive with respect nonexpansivity pole nonnegative Observe optimal solution positive real number problem 3.9 prove proximal point method respect to f sequence x}ken sequentially consistent stochastic convex feasibility strictly convex subgradient method subsets Suppose totally convex functions totally nonexpansive operators Tw(x vƒ(x weak accumulation points x}EN x*+¹ ΚΕΝ ΚΕΝ ΚΕΝ