Convex Optimization over Fixed Value Point Set of Quasi-Nonexpansive Random Operators on Hilbert Spaces

TL;DR

This paper introduces a new convex optimization framework over fixed point sets of quasi-nonexpansive random operators in Hilbert spaces, extending previous models to include centralized and distributed scenarios with proven convergence.

Contribution

It generalizes existing optimization frameworks to a broader class of operators and establishes convergence results for the proposed algorithms in Hilbert spaces.

Findings

The proposed algorithm converges almost surely.

The algorithm converges in mean square.

The framework encompasses previous models as special cases.

Abstract

In this paper, a new optimization framework is defined that includes the optimization framework recently proposed in [1]-[2] as a special case. The convex optimization in [1]-[2] includes centralized optimization and distributed optimization over random networks, so does the optimization defined here. It is shown that the proposed algorithm in [2] converges almost surely and in mean square to a solution of the optimization problem here under suitable assumptions.