# Open Journal of Mathematical Optimization

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.

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-${C}^{1,\alpha }$ 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.

Revised:
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
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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/

[1] Francis Bach Duality between subgradient and conditional gradient methods, SIAM J. Optim., Volume 26 (2015) no. 1, pp. 115-129 | DOI | Zbl

[2] Amir Beck First-Order Methods in Optimization, MOS-SIAM Series on Optimization, 25, Society for Industrial and Applied Mathematics, 2017 | DOI | Numdam

[3] Amir Beck; Nadav Hallak On the convergence to stationary points of deterministic and randomized feasible descent directions methods, SIAM J. Optim., Volume 30 (2020) no. 1, pp. 56-79 | DOI | Zbl

[4] Nicholas Boyd; Geoffrey Schiebinger; Benjamin Recht The alternating descent conditional gradient method for sparse inverse problems, SIAM J. Optim., Volume 27 (2017) no. 2, pp. 616-639 | DOI | Zbl

[5] Frank Clarke Optimisation and Nonsmooth Analysis, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics, 1990 | DOI

[6] Cyrille Combettes; Sebastian Pokutta Boosting Frank–Wolfe by chasing gradients, Proceedings of the 37th International Conference on Machine Learning (Proceedings of Machine Learning Research), Volume 119, PMLR, 2020, pp. 2111-2121

[7] Aris Daniilidis; Jérôme Malick Filling the gap between lower-${C}^{1}$ and lower-${C}^{2}$ functions, J. Convex Anal., Volume 12 (2005) no. 2, pp. 315-329 | Zbl

[8] Welington de Oliveira The ABC of DC programming, Set-Valued Var. Anal., Volume 28 (2020) no. 4, pp. 679-706 | DOI | Zbl

[9] Quentin Denoyelle; Vincent Duval; Gabriel Peyré; Emmanuel. Soubies The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy, Inverse Probl., Volume 36 (2020) no. 1, 014001, 42 pages | Zbl

[10] John C. Duchi; Peter L. Bartlett; Martin J. Wainwright Randomized smoothing for stochastic optimization, SIAM J. Optim., Volume 22 (2012) no. 2, pp. 674-701 | DOI | Zbl

[11] Marguerite Frank; Philip Wolfe An algorithm for quadratic programming, Nav. Res. Logist. Q., Volume 3 (1956) no. 1-2, pp. 95-110 | DOI

[12] Martin Jaggi Revisiting Frank–Wolfe: Projection-free sparse convex optimization, Proceedings of the 30th International Conference on Machine Learning (Proceedings of Machine Learning Research), Volume 28, PMLR, 2013, pp. 427-435

[13] Đ. Khuê Lê-Huu; Karteek Alahari Regularized Frank–Wolfe for dense CRFs: Generalizing mean field and beyond, Advances in Neural Information Processing Systems, Volume 34, Neural Information Processing Systems, 2021, pp. 1453-1467

[14] Hisashi Mine; Masao Fukushima A minimization method for the sum of a convex function and a continuously differentiable function, J. Optim. Theory Appl., Volume 33 (1981) no. 1, pp. 9-23 | DOI

[15] Yurii Nesterov Complexity bounds for primal-dual methods minimizing the model of objective function, Math. Program., Volume 171 (2018) no. 1-2, pp. 311-330 | DOI | Zbl

[16] Anton Osokin; Jean-Baptiste Alayrac; Isabella Lukasewitz; Puneet K. Dokania; Simon Lacoste-Julien Minding the gaps for block Frank–Wolfe optimization of structured SVMs, Proceedings of the 33rd International Conference of Machine Learning, ICML, 2016

[17] Jong-Shi Pang; Meisam Razaviyayn; Alberth Alvarado Computing B-stationary points of nonsmooth DC programs, Math. Oper. Res., Volume 42 (2017) no. 1, pp. 95-118 | DOI | Zbl

[18] Fabian Pedregosa; Geoffrey Negiar; Armin Askari; Martin Jaggi Linearly convergent Frank–Wolfe with backtracking line-search, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics (Proceedings of Machine Learning Research), Volume 108, PMLR, 2020, pp. 1-10

[19] Sathya N. Ravi; Maxwell D. Collins; Vikas Singh A deterministic nonsmooth Frank–Wolfe algorithm with coreset guarantees, INFORMS J. Optim., Volume 1 (2019) no. 2, pp. 120-142 | DOI

[20] R. Tyrrell Rockafellar Favorable classes of lipschitz continuous functions in subgradient optimization, Progress in Nondifferentiable Optimization (IIASA Collaborative Proceedings Series), International Institute of Applied Systems Analysis, 1982, pp. 125-144 | Zbl

[21] R. Tyrrell Rockafellar Generalized subgradients in mathematical programming, Mathematical Programming The State of the Art, XIth International Symposium on Mathematical Programming, Bonn, Germany, August 23-27, Springer, 1982, pp. 23-27

[22] R. Tyrrell Rockafellar; Roger J.-B. Wets Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer, 2009

[23] Jonathan E. Spingarn Submonotone subdifferentials of Lipschitz functions, Trans. Am. Math. Soc., Volume 264 (1981), pp. 77-89 | DOI | Zbl

[24] Kiran Koshy Thekumparampil; Prateek Jain; Praneeth Netrapalli; Sewoong Oh Projection efficient subgradient method and optimal nonsmooth Frank–Wolfe method, Advances in Neural Information Processing Systems 33, Curran Associates, Inc., 2020, pp. 12211-12224

[25] Douglas J. White Extension of the Frank–Wolfe algorithm to concave nondifferentiable objective functions, J. Optim. Theory Appl., Volume 78 (1993) no. 2, pp. 283-301 | DOI | Zbl

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