An extension of the proximal point algorithm beyond convexity.

Area of Science:

• Optimization Theory
• Convex Analysis
• Applied Mathematics

Background:

• Convexity is crucial for optimization algorithms like the proximal point algorithm.
• Existing convexity notions have limitations in capturing certain function properties relevant to optimization.

Purpose of the Study:

• Introduce and investigate a novel generalized convexity concept: prox-convexity.
• Analyze the properties of the proximity operator for prox-convex functions.
• Determine the convergence of the proximal point algorithm under prox-convexity.

Main Methods:

• Definition of prox-convexity and its properties.
• Investigation of the proximity operator's behavior (single-valued, firmly nonexpansive).
• Comparison of prox-convexity with existing classes (quasiconvex, weakly convex, DC functions).
• Analysis of the proximal point algorithm's convergence for prox-convex functions.

Main Results:

• Prox-convexity is a new function class, distinct from quasiconvex, weakly convex, and DC functions.
• The proximity operator of a prox-convex function is single-valued and firmly nonexpansive.
• The proximal point algorithm converges for proper lower semicontinuous prox-convex functions.

Conclusions:

• Prox-convexity offers a valuable generalization of convexity in optimization.
• The established convergence properties extend the applicability of proximal algorithms.
• This work bridges existing convexity concepts and introduces a new framework for optimization research.