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

TL;DR

The paper tackles nonsmooth, nonconvex composite optimization on Hadamard manifolds by formulating problems as $f(x)=g_1(x)+g_2(x)-h(x)$ with $h$ convex, $g_2$ having a Lipschitz gradient, and $g_1$ lsc. It develops two Riemannian proximal point variants: a standard version requiring the Lipschitz constant $L$ and an adaptive version (Adap-RPPM) that does not rely on $L$, both achieving $O(\epsilon^{-2})$ iteration complexity. The authors establish convergence results, including KL-based global convergence under suitable assumptions, and illustrate the methods on log-determinant DCG problems with numerical experiments demonstrating scalability to high dimensions and efficiency relative to baseline methods. The work advances proximal methods on manifolds, offers practical algorithms for DCG-type problems in non-Euclidean settings, and suggests directions for acceleration and broader Hölder-type analyses.

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.

Figures (3)

• Figure 1: Comparison of Adap-RPPM and DCPPA in terms of CPU run time for the $f_1$ problem.
• Figure 2: Comparison of Adap-RPPM and DCPPA in terms of CPU run time for the $f_2$ problem.
• Figure 3: Performance comparison of Adap-RPPM, DCA, and DCPPA for the $f_3$ problem.

Theorems & Definitions (24)

• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4