Improved convergence rates for the multiobjective Frank-Wolfe method

TL;DR

This paper improves the theoretical convergence rates of the multiobjective Frank-Wolfe algorithm under various assumptions, including strong convexity and uniform convexity of the feasible set, with practical implications for convex constrained multiobjective optimization.

Contribution

It establishes new, faster convergence rates for the Frank-Wolfe method in multiobjective convex optimization under different geometric and functional assumptions.

Findings

Linear convergence under strong convexity and interior limit points.

Convergence rates of (1/k^{q/(q-1)}) for strongly convex objectives with ((1/k^{q/(q-1)})) rate for certain convex sets.

Enhanced convergence rates based on optimality measures.

Abstract

This paper analyzes the convergence rates of the {\it 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. In terms of the objective function values, we firstly show that if the objective function is strongly convex and the limit point of the sequence generated by the method lies in the relative interior of the feasible set, then the algorithm achieves a linear convergence rate. Next, we focus on a special class of problems where the feasible constraint set is $(\alpha,q)$-uniformly convex for some $\alpha >0$ and $q \geq 2$, including, in particular, \(\ell_p\)-balls for all $p>1$. In this context, we prove that the method attains: (i) a rate of $\mathcal{O}(1/k^\frac{q}{q-1})$ when the objective function is strongly convex; and (ii) a linear rate (if $q=2$) or a rate of $\mathcal{O}(1/k^{\frac{q}{q-2}})$ (if $q>2$) under an additional assumption, which always holds if the feasible set does not contain an unconstrained weak Pareto point. We also discuss enhanced convergence rates for the algorithm in terms of an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.