Splitting proximal point algorithms for the sum of prox-convex functions

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: 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 assumptions, 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.