Solving Low-Rank Semidefinite Programs via Manifold Optimization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916710853902336 |
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| author | Wang, Jie Hu, Liangbing |
| author_facet | Wang, Jie Hu, Liangbing |
| contents | We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented Lagrangian method (ALM) and the Burer-Monteiro factorization, and features the self-adaptive strategies for updating the factorization size and the penalty parameter. We establish global convergence of the inexact ALM, despite the non-convexity brought by the Burer-Monteiro factorization. We further provide a practical algorithm building on the inexact ALM, and along with the algorithm we release an open-source SDP solver ManiSDP. Comprehensive numerical experiments demonstrate that ManiSDP achieves state-of-the-art in terms of efficiency, accuracy, and scalability, and is faster than several advanced SDP solvers (MOSEK, SDPLR, SDPNAL+, STRIDE) by up to orders of magnitudes on a variety of linear SDPs. The largest SDP solved by ManiSDP (in about 8.5 hours with maximal KKT residue 3.5e-13) is the second-order moment relaxation of a binary quadratic program with 120 variables, which has matrix dimension 7261 and contains 17,869,161 affine constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solving Low-Rank Semidefinite Programs via Manifold Optimization Wang, Jie Hu, Liangbing Optimization and Control Primary, 90C22, Secondary, 90C23, 90C30 We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented Lagrangian method (ALM) and the Burer-Monteiro factorization, and features the self-adaptive strategies for updating the factorization size and the penalty parameter. We establish global convergence of the inexact ALM, despite the non-convexity brought by the Burer-Monteiro factorization. We further provide a practical algorithm building on the inexact ALM, and along with the algorithm we release an open-source SDP solver ManiSDP. Comprehensive numerical experiments demonstrate that ManiSDP achieves state-of-the-art in terms of efficiency, accuracy, and scalability, and is faster than several advanced SDP solvers (MOSEK, SDPLR, SDPNAL+, STRIDE) by up to orders of magnitudes on a variety of linear SDPs. The largest SDP solved by ManiSDP (in about 8.5 hours with maximal KKT residue 3.5e-13) is the second-order moment relaxation of a binary quadratic program with 120 variables, which has matrix dimension 7261 and contains 17,869,161 affine constraints. |
| title | Solving Low-Rank Semidefinite Programs via Manifold Optimization |
| topic | Optimization and Control Primary, 90C22, Secondary, 90C23, 90C30 |
| url | https://arxiv.org/abs/2303.01722 |