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Journal of Convex Analysis 10 (2003), No. 1, 035--061
Copyright Heldermann Verlag 2003
On Uniform Convexity, Total Convexity and Convergence of the Proximal Point and Outer
Bregman Projection Algorithm in Banach Spaces
Dan Butnariu
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel,
dbutnaru@math.haifa.ac.il
Alfredo N. Iusem
Institute of Pure and Applied Mathematics (IMPA), Estrada Dona Castorina 110,
Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil,
iusp@impa.br
Constantin Zalinescu
Faculty of Mathematics, "Al. I. Cuza" University, 6600 Iasi, Romania,
zalinesc@uaic.ro
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.
Keywords: Uniform convexity at a point, total convexity at a point, uniform
convexity on bounded sets, sequential consistency, generalized proximal point
algorithm for optimization, outer Bregman projection algorithm for feasibily.
MSC 2000: 90C25, 90C48, 26B25, 49J52, 46N10, 46N20, 90C30.
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