Last-Iterate Convergence of Frank--Wolfe for Nonlinear Variational Inequalities on Polytopes

Abstract

We consider the Frank--Wolfe algorithm for solving variational inequalities over compact polytopes under a monotone $C^1$ operator $\Phi$. The linear minimization oracle returns vertices, and we assume a coercivity condition on the Jacobian of $\Phi$ at a solution. We prove last-iterate convergence to the unique solution at rate $O(k^{-\min\{\alpha,1\}})$ for step sizes $\gamma_k=\alpha/(k+\beta)$ with $\beta\geq\alpha>0$. The bound holds uniformly over starting points and vertex tie-breaking. In particular, both $1/(k+1)$ and $2/(k+2)$ give $O(k^{-1})$ distance. A matching lower bound shows that this distance guarantee is worst-case order-optimal among prescribed admissible step-size schedules. We also obtain rates $O(k^{-\min\{\alpha,1\}})$ for the Frank--Wolfe gap and $O(k^{-\min\{\alpha,2\}})$ for the dual gap. These yield $O(\varepsilon^{-1})$ and $O(\varepsilon^{-1/2})$ calls to each oracle for sustained gap accuracy $\varepsilon$ when $\alpha\geq1$ and $\alpha\geq2$, respectively, with instance-dependent constants. We also treat more general prescribed step-size schedules.