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On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces. (English) Zbl 1091.90078
Summary: In this paper we study and compare the notions of uniform convexity of functions at a point and on bounded sets with the notions of total convexity at a point and sequential consistency of functions, respectively. We establish connections between these concepts of strict convexity in infinite dimensional settings and use the connections in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces.

MSC:
90C48 Programming in abstract spaces
90C26 Nonconvex programming, global optimization
46B20 Geometry and structure of normed linear spaces
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
49J27 Existence theories for problems in abstract spaces
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