A generalized conditional gradient method for multiobjective composite optimization problems

TL;DR

This paper introduces a generalized conditional gradient method for multiobjective composite optimization, analyzing its convergence and complexity with various step size strategies, supported by numerical experiments.

Contribution

It proposes a novel generalized conditional gradient method tailored for multiobjective composite problems, with comprehensive convergence analysis and practical performance evaluation.

Findings

Convergence established for the proposed method.

Iteration complexity bounds derived under different assumptions.

Numerical experiments demonstrate practical effectiveness.

Abstract

This article deals with multiobjective composite optimization problems that consist of simultaneously minimizing several objective functions, each of which is composed of a combination of smooth and non-smooth functions. To tackle these problems, we propose a generalized version of the conditional gradient method, also known as Frank-Wolfe method. The method is analyzed with three step size strategies, including Armijo-type, adaptive, and diminishing step sizes. We establish asymptotic convergence properties and iteration-complexity bounds, with and without convexity assumptions on the objective functions. Numerical experiments illustrating the practical behavior of the methods are presented.