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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Université de Montpellier
2023-01-01
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| Series: | Open Journal of Mathematical Optimization |
| Subjects: | |
| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.21/ |
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| Summary: | 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. |
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| ISSN: | 2777-5860 |