A conditional gradient homotopy method with applications to Semidefinite Programming
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912209529995264 |
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| author | Dvurechensky, Pavel Iommazzo, Gabriele Shtern, Shimrit Staudigl, Mathias |
| author_facet | Dvurechensky, Pavel Iommazzo, Gabriele Shtern, Shimrit Staudigl, Mathias |
| contents | We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03101 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A conditional gradient homotopy method with applications to Semidefinite Programming Dvurechensky, Pavel Iommazzo, Gabriele Shtern, Shimrit Staudigl, Mathias Optimization and Control Machine Learning 90C22, 90C25 We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method. |
| title | A conditional gradient homotopy method with applications to Semidefinite Programming |
| topic | Optimization and Control Machine Learning 90C22, 90C25 |
| url | https://arxiv.org/abs/2207.03101 |