Stochastic Approximation Method for Fixed Point Problems

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

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.

DOI: 10.4236/am.2012.312A293                

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

Conflicts of Interest

The authors declare no conflicts of interest.

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