Stochastic Approximation Method for Fixed Point Problems - Applied Mathematics

AM > Vol.3 No.12, December 2012

Stochastic Approximation Method for Fixed Point Problems ()

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DOI: 10.4236/am.2012.312A293 3,770 Downloads 6,710 Views Citations

Author(s)

Ya. I. Alber, C. E. Chidume, Jinlu Li

Affiliation(s)

Department of Mathematical Sciences, Shawnee State University, Portsmouth, USA.
Department of Mathematics, The Technion-Israel Institute of Technology, Haifa, Israel.
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.

KEYWORDS

Hilbert Spaces; Stochastic Approximation Algorithm; Weakly Contractive Operators; Nonexpansive Operators; Fixed Points; Convergence in Mean Square; Convergence Almost Sure (a.s.); Nonasymptotic Estimates of Convergence Rate

Share and Cite:

Y. Alber, C. Chidume and J. Li, "Stochastic Approximation Method for Fixed Point Problems," Applied Mathematics, Vol. 3 No. 12A, 2012, pp. 2123-2132. doi: 10.4236/am.2012.312A293.

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