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arXiv · 2507.17272

Frank-Wolfe algorithm for star-convex functions

R. Diaz Millan·Orizon Pereira Ferreira·Julien Ugon
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Abstract

We study the Frank-Wolfe algorithm for minimizing a differentiable function with Lipschitz continuous gradient over a compact convex set. To extend classical complexity bounds to certain non-convex functions, we focus on the class of \emph{star-convex functions}, which retain essential geometric properties despite the lack of convexity. We establish iteration-complexity bounds of $\mathcal{O}(1/k)$ for both the objective values and the duality gap under star-convexity, using diminishing, Armijo-type, and Lipschitz-based stepsize rules. Notably, the diminishing and Armijo strategies do not require prior knowledge of Lipschitz or curvature constants. These results demonstrate that the Frank-Wolfe method preserves optimal complexity guarantees beyond the convex setting.

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R. Diaz Millan, Orizon Pereira Ferreira, Julien Ugon. 2025-07-23. Frank-Wolfe algorithm for star-convex functions. https://arxiv.org/abs/2507.17272

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