Splitting Proximal Point Algorithms for the Sum of Prox-Convex Functions

TL;DR

This paper extends splitting proximal point algorithms to minimize sums of prox-convex functions, proving convergence and demonstrating efficiency through numerical experiments.

Contribution

It introduces deterministic and stochastic variants of splitting proximal point algorithms for prox-convex functions, extending previous convex-focused methods.

Findings

Both algorithms converge globally under standard stepsize assumptions.

The stochastic variant achieves almost sure convergence via supermartingale theory.

Numerical experiments confirm the methods' efficiency on nonconvex quadratic functions.

Abstract

This paper addresses the minimization of a finite sum of prox-convex functions under Lipschitz continuity of each component. We propose two variants of the splitting proximal point algorithms proposed in \cite{Bacak,Bertsekas}: one deterministic with a fixed update order, and one stochastic with random sampling, and we extend them from convex to prox-convex functions. We prove global convergence for both methods under standard stepsize a\-ssump\-tions, with almost sure convergence for the stochastic variant via supermartingale theory. Numerical experiments with nonconvex quadratic functions illustrate the efficiency of the proposed methods and support the theoretical results.