Proximal Point Algorithm for Quasi-convex Minimization Problems in metric spaces

TL;DR

This paper extends the proximal point algorithm to quasi-convex minimization in nonpositive curvature metric spaces, proving convergence to critical points and minima, thus broadening its applicability beyond Hadamard manifolds.

Contribution

It introduces convergence results for the proximal point algorithm in general nonpositive curvature metric spaces, extending previous work in Hadamard manifolds and CAT(0) spaces.

Findings

Sequence $ riangle$-converges to a critical point

Strong convergence to a minimum under additional conditions

Extends proximal point algorithm results to broader metric spaces

Abstract

In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove $\Delta$-convergence of the generated sequence to a critical point (which is defined in the text) of an objective convex, proper and lower semicontinuous function with at least a minimum point as well as some strong convergence results to a minimum point with some additional conditions. The results extend the recent results of the proximal point algorithm in Hadamard manifolds and CAT(0) spaces.