An extension of the proximal point algorithm beyond convexity

TL;DR

This paper introduces prox-convexity, a new generalized convexity concept, and demonstrates that the classical proximal point algorithm converges under this broader condition, extending its applicability beyond convex functions.

Contribution

The paper defines prox-convexity, explores its properties, and proves convergence of the proximal point algorithm for prox-convex functions, expanding the scope of optimization methods.

Findings

Prox-convex functions have single-valued, firmly nonexpansive proximity operators.

Classical proximal point algorithm converges for prox-convex functions.

Examples include quasiconvex, weakly convex, and DC functions that are prox-convex.

Abstract

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.