A Proximal Gradient Framework for Composite Multiobjective Optimization on Riemannian Manifolds

TL;DR

This paper introduces a Riemannian multiobjective proximal gradient method that directly optimizes vector objectives on manifolds, ensuring convergence and efficiency improvements over existing subgradient methods.

Contribution

It develops a novel framework for multiobjective optimization on Riemannian manifolds that directly handles vector objectives and provides convergence guarantees.

Findings

The method achieves an $ ext{O}(1/k)$ convergence rate.

The inexact and trust-region variants improve practicality and complexity.

Numerical experiments show superior performance over subgradient baselines.

Abstract

This paper proposes a Riemannian Multiobjective Proximal Gradient Method (RMPGM) for composite optimization problems on manifolds. Unlike scalarization-based approaches, the proposed framework directly handles vector-valued objectives and establishes global convergence to Pareto stationary points, together with an $\mathcal{O}(1/k)$ convergence rate. We further develop two variants to enhance practicality and performance: an inexact RMPGM that allows controlled inexactness in solving subproblems, and a trust-region RMPGM that adaptively adjusts the penalty parameter and achieves an $\mathcal{O}(\epsilon^{-2}) $iteration complexity. Numerical experiments demonstrate that the proposed methods are consistently outperform subgradient-based baselines.