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A Nonsmooth Frank–Wolfe Algorithm Through a Dual Cutting-Plane Approach

Author

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  • Guilherme Mazanti

    (Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes
    Fédération de Mathématiques de CentraleSupélec)

  • Thibault Moquet

    (Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes
    Fédération de Mathématiques de CentraleSupélec)

  • Laurent Pfeiffer

    (Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes
    Fédération de Mathématiques de CentraleSupélec)

Abstract

An extension of the Frank–Wolfe Algorithm (FWA), also known as Conditional Gradient algorithm, is proposed. In its standard form, the FWA allows to solve constrained optimization problems involving $$\beta $$ β -smooth cost functions, calling at each iteration a Linear Minimization Oracle. More specifically, the oracle solves a problem obtained by linearization of the original cost function. The algorithm designed and investigated in this article, named Dualized Level-Set (DLS) algorithm, extends the FWA and allows to address a class of nonsmooth costs, involving in particular support functions. The key idea behind the construction of the DLS method is a general interpretation of the FWA as a cutting-plane algorithm, from the dual point of view. The DLS algorithm essentially results from a dualization of a specific cutting-plane algorithm, based on projections on some level sets. The DLS algorithm generates a sequence of primal-dual candidates, and we prove that the corresponding primal-dual gap converges with a rate of $$O(1/\sqrt{t})$$ O ( 1 / t ) .

Suggested Citation

  • Guilherme Mazanti & Thibault Moquet & Laurent Pfeiffer, 2025. "A Nonsmooth Frank–Wolfe Algorithm Through a Dual Cutting-Plane Approach," Journal of Optimization Theory and Applications, Springer, vol. 207(2), pages 1-49, November.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:2:d:10.1007_s10957-025-02752-y
    DOI: 10.1007/s10957-025-02752-y
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    References listed on IDEAS

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    1. 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.
    2. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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