Sparse Quadratically Constrained Quadratic Programming via Semismooth Newton Method
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918156064260096 |
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| author | Li, Shuai Zhou, Shenglong Luo, Ziyan |
| author_facet | Li, Shuai Zhou, Shenglong Luo, Ziyan |
| contents | Quadratically constrained quadratic programming (QCQP) has long been recognized as a computationally challenging problem, particularly in large-scale or high-dimensional settings where solving it directly becomes intractable. The complexity further escalates when a sparsity constraint is involved, giving rise to the problem of sparse QCQP (SQCQP), which makes conventional solution methods even less effective. Existing approaches for solving SQCQP typically rely on mixed-integer programming formulations, relaxation techniques, or greedy heuristics but often suffer from computational inefficiency and limited accuracy. In this work, we introduce a novel paradigm by designing an efficient algorithm that directly addresses SQCQP. To be more specific, we introduce P-stationarity to establish first- and second-order optimality conditions of the original problem, leading to a system of nonlinear equations whose generalized Jacobian is proven to be nonsingular under mild assumptions. Most importantly, these equations facilitate the development of a semismooth Newton-type method that exhibits significantly low computational complexity due to the sparsity constraint and achieves a locally quadratic convergence rate. Finally, extensive numerical experiments validate the accuracy and computational efficiency of the algorithm compared to several established solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse Quadratically Constrained Quadratic Programming via Semismooth Newton Method Li, Shuai Zhou, Shenglong Luo, Ziyan Optimization and Control Quadratically constrained quadratic programming (QCQP) has long been recognized as a computationally challenging problem, particularly in large-scale or high-dimensional settings where solving it directly becomes intractable. The complexity further escalates when a sparsity constraint is involved, giving rise to the problem of sparse QCQP (SQCQP), which makes conventional solution methods even less effective. Existing approaches for solving SQCQP typically rely on mixed-integer programming formulations, relaxation techniques, or greedy heuristics but often suffer from computational inefficiency and limited accuracy. In this work, we introduce a novel paradigm by designing an efficient algorithm that directly addresses SQCQP. To be more specific, we introduce P-stationarity to establish first- and second-order optimality conditions of the original problem, leading to a system of nonlinear equations whose generalized Jacobian is proven to be nonsingular under mild assumptions. Most importantly, these equations facilitate the development of a semismooth Newton-type method that exhibits significantly low computational complexity due to the sparsity constraint and achieves a locally quadratic convergence rate. Finally, extensive numerical experiments validate the accuracy and computational efficiency of the algorithm compared to several established solvers. |
| title | Sparse Quadratically Constrained Quadratic Programming via Semismooth Newton Method |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2503.15109 |