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

We consider the Frank–Wolfe algorithm for solving variational inequalities over compact polytopes under a monotone $C^1$ operator $\mathrm{\Phi }$. The linear minimization oracle returns vertices, and we assume a coercivity condition on the Jacobian of $\mathrm{\Phi }$ at a solution. We prove last-iterate convergence to the unique solution at rate $O(k^{-\min \{\alpha ,1\}})$ for step sizes ${\gamma }_k=\alpha \mathrm{/}(k+\beta )$ with $\beta \ge \alpha >0$. The bound holds uniformly over starting points and vertex tie-breaking. In particular, both $1\mathrm{/}(k+1)$ and $2\mathrm{/}(k+2)$ 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 $O({\epsilon }^{-1})$ and $O({\epsilon }^{-1\mathrm{/}2})$ calls to each oracle for sustained gap accuracy $\epsilon$ when $\alpha \ge 1$ and $\alpha \ge 2$, respectively, with instance-dependent constants. We also treat more general prescribed step-size schedules.