The difference of convex algorithm on Hadamard manifolds

TL;DR

This paper introduces a Riemannian version of the difference of convex algorithm (DCA) for optimization on Hadamard manifolds, proving its theoretical properties and demonstrating its effectiveness through numerical experiments.

Contribution

It develops a Riemannian DCA for Hadamard manifolds, establishing equivalence with classical DCA and analyzing convergence and duality properties.

Findings

The Riemannian DCA is well-defined under mild conditions.

Every cluster point of the sequence is a critical point.

Numerical experiments confirm the algorithm's effectiveness.

Abstract

In this paper, we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. We establish the equivalence between the classical and simplified Riemannian versions of the DCA. We also prove that, under mild assumptions, the Riemannian version of the DCA is well-defined, and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Additionally, we establish some duality relations between the DC problem and its dual. To illustrate the effectiveness of the algorithm, we present some numerical experiments.