Open Journal of Mathematical Optimization

Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems
Welington de Oliveira1
1 Mines Paris, Université PSL, Centre de Mathématiques Appliquées (CMA), 06904 Sophia Antipolis, France
Open Journal of Mathematical Optimization, Volume 4 (2023), article no. 2, 10 p.

Frank and Wolfe’s celebrated conditional gradient method is a well-known tool for solving smooth optimization problems for which minimizing a linear function over the feasible set is computationally cheap. However, when the objective function is nonsmooth, the method may fail to compute a stationary point. In this work, we show that the Frank–Wolfe algorithm can be employed to compute Clarke-stationary points for nonconvex and nonsmooth optimization problems consisting of minimizing upper-C 1,α functions over convex and compact sets. Furthermore, under more restrictive assumptions, we propose a new algorithm variant with stronger stationarity guarantees, namely directional stationarity and even local optimality.

Received:
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Accepted:
Published online:
DOI: 10.5802/ojmo.21
Keywords: Nonsmooth Optimization, Nonconvex Optimization, Frank–Wolfe Algorithm
Welington de Oliveira 1

1 Mines Paris, Université PSL, Centre de Mathématiques Appliquées (CMA), 06904 Sophia Antipolis, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Welington de Oliveira. Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems. Open Journal of Mathematical Optimization, Volume 4 (2023), article  no. 2, 10 p. doi : 10.5802/ojmo.21. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.21/

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