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Zbl 1101.90083
Kamimura, Shoji; Takahashi, Wataru
Strong convergence of a proximal-type algorithm in a Banach space. (English)
[J] SIAM J. Optim. 13, No. 3, 938-945 (2003). ISSN 1052-6234; ISSN 1095-7189/e

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)], {\it M. V. Solodov} and {\it 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 2000:: *90C48 Programming in abstract spaces
47H05 Monotone operators (with respect to duality)
47J25 Methods for solving nonlinear operator equations (general)

Keywords: proximal point algorithm; maximal monotone operator; Banach space; strong convergence

Citations: Zbl 0971.90062

Cited in: Zbl 1201.49033 Zbl 1200.47074 Zbl 1129.49005 Zbl 1125.47047

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