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Article Dans Une Revue Mathematics of Operations Research Année : 2023

The Iterates of the Frank-Wolfe Algorithm May Not Converge

Résumé

The Frank-Wolfe algorithm is a popular method for minimizing a smooth convex function $f$ over a compact convex set $\mathcal{C}$. While many convergence results have been derived in terms of function values, hardly nothing is known about the convergence behavior of the sequence of iterates $(x_t)_{t\in\mathbb{N}}$. Under the usual assumptions, we design several counterexamples to the convergence of $(x_t)_{t\in\mathbb{N}}$, where $f$ is $d$-time continuously differentiable, $d\geq2$, and $f(x_t)\to\min_\mathcal{C}f$. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies. We do not assume \emph{misspecification} of the linear minimization oracle and our results thus hold regardless of the points it returns, demonstrating the fundamental pathologies in the convergence behavior of $(x_t)_{t\in\mathbb{N}}$.
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Dates et versions

hal-03579585 , version 1 (18-02-2022)
hal-03579585 , version 2 (18-07-2024)

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Jérôme Bolte, Cyrille W. Combettes, Edouard Pauwels. The Iterates of the Frank-Wolfe Algorithm May Not Converge. Mathematics of Operations Research, inPress, ⟨10.1287/moor.2022.0057⟩. ⟨hal-03579585v2⟩
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