Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes
Optimization and Control
2025-09-11 v2 Machine Learning
Abstract
We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are -Polyak-{\L}ojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.
Cite
@article{arxiv.2505.11259,
title = {Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes},
author = {Gabriele Iommazzo and David Martínez-Rubio and Francisco Criado and Elias Wirth and Sebastian Pokutta},
journal= {arXiv preprint arXiv:2505.11259},
year = {2025}
}