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arXiv · 2505.00221

Strongly Convex Maximization via the Frank-Wolfe Algorithm with the Kurdyka-{\L}ojasiewicz Inequality

Fatih Selim Aktas·Christian Kroer
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Abstract

We study the convergence properties of the 'greedy' Frank-Wolfe algorithm with a unit step size, for a convex maximization problem over a compact set. We assume the function satisfies smoothness and strong convexity. These assumptions together with the Kurdyka-{\L}ojasiewicz (KL) property, allow us to derive global asymptotic convergence for the sequence generated by the algorithm. Furthermore, we also derive a convergence rate that depends on the geometric properties of the problem. To illustrate the implications of the convergence result obtained, we prove a new convergence result for a sparse principal component analysis algorithm, propose a convergent reweighted $\ell_1$ minimization algorithm for compressed sensing, and design a new algorithm for the semidefinite relaxation of the Max-Cut problem.

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Fatih Selim Aktas, Christian Kroer. 2025-04-30. Strongly Convex Maximization via the Frank-Wolfe Algorithm with the Kurdyka-{\L}ojasiewicz Inequality. https://arxiv.org/abs/2505.00221

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