Convergence of the Frank-Wolfe Algorithm for Monotone Variational Inequalities

TL;DR

This paper analyzes the convergence properties of the Frank-Wolfe algorithm for monotone variational inequalities using dynamical systems theory, establishing asymptotic convergence and rates.

Contribution

It introduces a continuous-time analysis of the Frank-Wolfe algorithm, proving convergence and addressing Hammond's conjecture on generalized fictitious play.

Findings

Cluster points of iterates are solutions.

Distance to solution set converges to zero.

In strongly monotone case, iterates converge to the unique solution.

Abstract

We consider the Frank-Wolfe algorithm for solving variational inequalities over compact, convex sets under a monotone $C^1$ operator and vanishing, nonsummable step sizes. We introduce a continuous-time interpolation of the discrete iteration and use tools from dynamical systems theory to analyze its asymptotic behavior. This allows us to derive convergence results for the original discrete algorithm. Consequently, every cluster point of the iterates is a solution of the underlying variational inequality, the distance from the iterates to the solution set converges to zero, and the Frank-Wolfe gap vanishes asymptotically. In the strongly monotone case, the solution is unique and the iterates converge to it. In particular, this proves Hammond's conjecture on the convergence of generalized fictitious play. We also discuss rates of convergence and under what assumptions rates can be shown.