Random Function Iterations for Consistent Stochastic Feasibility

• Published in: Numerical functional analysis and optimization Vol. 40; no. 4; pp. 386 - 420
• Main Authors: Hermer, Neal, Luke, D. Russell, Sturm, Anja
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 12.03.2019; Taylor & Francis Ltd
• Subjects: Algorithms; Averaged mappings; Convergence; Feasibility studies; geometric convergence of measures; iterated random functions; linear convergence in expectation; linear regularity; Markov chains; metric subregularity; nonexpansive mappings; paracontractions; Primary 60J05; Secondary 49J53, 65K05; stochastic feasibility; stochastic fixed point problem
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630563.2018.1535507

We study the convergence of iterated random functions for stochastic feasibility in the consistent case (in the sense of Butnariu and Flåm [Numer. Funct. Anal. Optimiz., 1995]) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point mappings. The iterations are Markov chains and, for the purposes of this study, convergence is understood in very restrictive terms. We show that sufficient conditions for geometric (linear) convergence in expectation of stochastic projection algorithms presented in Nedić [Math. Program, 2011], are in fact necessary for geometric (linear) convergence in expectation more generally of iterated random functions.