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Strong convergence of a proximal-type algorithm in a Banach space. (English) Zbl 1101.90083
Summary: We study strong convergence of the proximal point algorithm. It is known that the proximal point algorithm converges weakly to a solution of a maximal monotone operator, but it fails to converge strongly. Then, in [Math. Program. 87, No. 1(A), 189–202 (2000; Zbl 0971.90062)], M. V. Solodov and B. F. Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for it in Hilbert spaces. Our purpose is to extend Solodov and Svaiter’s result to more general Banach spaces. Using this, we consider the problem of finding a minimizer of a convex function.

MSC:
90C48Programming in abstract spaces
47H05Monotone operators (with respect to duality) and generalizations
47J25Iterative procedures (nonlinear operator equations)