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The Iterates of the Frank-Wolfe Algorithm May Not Converge

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  • Bolte, Jérôme
  • Combettes, Cyrille
  • Pauwels, Edouard

Abstract

The Frank-Wolfe algorithm is a popular method for minimizing a smooth convex function f over a compact convex set 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 (xt)t2N. Under the usual assumptions, we design several counterexamples to the convergence of (xt)t2N, where f is d-time continuously differentiable, d > 2, and f(xt) ---> minC f. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies. We do not assume 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 (xt)t2N.

Suggested Citation

  • Bolte, Jérôme & Combettes, Cyrille & Pauwels, Edouard, 2022. "The Iterates of the Frank-Wolfe Algorithm May Not Converge," TSE Working Papers 22-1311, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:126672
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
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