Improved convergence rates for the multiobjective Frank-Wolfe method

Abstract

This paper analyzes the convergence rates of the Frank–Wolfe method for solving convex constrained multiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. Notably, we demonstrate that the method can achieve linear convergence rates in terms of a merit function whenever the objectives are strongly convex and the limit point is in the relative interior of the feasible set, or when the feasible set is strongly convex and it does not contain an unconstrained weak Pareto point. Moreover, improved sublinear convergence rates can also be obtained in other scenarios where the feasible set is uniformly convex. Additionally, we explore enhanced convergence rates with respect to an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.