An Adaptive Proximal Point Method for Nonsmooth and Nonconvex Optimization on Hadamard Manifolds

TL;DR

This paper introduces an adaptive proximal point method for complex nonsmooth, nonconvex optimization problems on Hadamard manifolds, with convergence analysis and practical numerical demonstrations.

Contribution

It develops two variants of an adaptive proximal point algorithm tailored for nonsmooth nonconvex problems on Riemannian manifolds, expanding applicability and providing convergence guarantees.

Findings

Both methods converge under specified conditions.

Numerical experiments demonstrate effectiveness.

The second variant works without Lipschitz constant knowledge.

Abstract

This paper addresses a class of nonsmooth and nonconvex optimization problems defined on complete Riemannian manifolds. The objective function has a composite structure, combining convex, differentiable, and lower semicontinuous terms, thereby generalizing the classical framework of difference-of-convex programming. Motivated by recent advances in proximal point methods in Euclidean and Riemannian settings, we propose two variants: one that uses the Lipschitz constant of the gradient of the smooth part, suitable when this parameter is accessible, and another that dispenses with such knowledge, expanding its applicability. We analyze the complexity of both approaches, establish their convergence, and illustrate their effectiveness through numerical experiments.