Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems
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...
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| Format: | Article |
| Language: | English |
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Université de Montpellier
2023-01-01
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| Series: | Open Journal of Mathematical Optimization |
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| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.21/ |
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| author | de Oliveira, Welington |
| author_facet | de Oliveira, Welington |
| author_sort | de Oliveira, Welington |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-7ec464c636914666bfbf39b3fc330ce3 |
| institution | Kabale University |
| issn | 2777-5860 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Université de Montpellier |
| record_format | Article |
| series | Open Journal of Mathematical Optimization |
| spelling | doaj-art-7ec464c636914666bfbf39b3fc330ce32025-02-07T14:02:56ZengUniversité de MontpellierOpen Journal of Mathematical Optimization2777-58602023-01-01411010.5802/ojmo.2110.5802/ojmo.21Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problemsde Oliveira, Welington0Mines Paris, Université PSL, Centre de Mathématiques Appliquées (CMA), 06904 Sophia Antipolis, FranceFrank 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.https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.21/Nonsmooth OptimizationNonconvex OptimizationFrank–Wolfe Algorithm |
| spellingShingle | de Oliveira, Welington Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems Open Journal of Mathematical Optimization Nonsmooth Optimization Nonconvex Optimization Frank–Wolfe Algorithm |
| title | Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| title_full | Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| title_fullStr | Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| title_full_unstemmed | Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| title_short | Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| title_sort | short paper a note on the frank wolfe algorithm for a class of nonconvex and nonsmooth optimization problems |
| topic | Nonsmooth Optimization Nonconvex Optimization Frank–Wolfe Algorithm |
| url | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.21/ |
| work_keys_str_mv | AT deoliveirawelington shortpaperanoteonthefrankwolfealgorithmforaclassofnonconvexandnonsmoothoptimizationproblems |