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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.21810 |
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| _version_ | 1866912455155777536 |
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| author | Hieu, Vu Trung Iusem, Alfredo Noel Schmölling, Paul Hugo Takeda, Akiko |
| author_facet | Hieu, Vu Trung Iusem, Alfredo Noel Schmölling, Paul Hugo Takeda, Akiko |
| contents | By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper points out that, generically, this polynomial system has finitely many complex zeroes. In such a case, we use symbolic computation techniques to compute a univariate representation of the solution set. Consequently, univariate representations of special solutions, such as least-norm and sparse solutions, are obtained. After that, enumerating solutions boils down to solving problems governed by univariate polynomials. We also provide some experiments on small-scale problems with worst-case scenarios. At the end of the paper, we propose a method for computing approximate solutions to copositive polynomial complementarity problems that may have infinitely many solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Univariate representations of solutions to generic polynomial complementarity problems Hieu, Vu Trung Iusem, Alfredo Noel Schmölling, Paul Hugo Takeda, Akiko Optimization and Control By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper points out that, generically, this polynomial system has finitely many complex zeroes. In such a case, we use symbolic computation techniques to compute a univariate representation of the solution set. Consequently, univariate representations of special solutions, such as least-norm and sparse solutions, are obtained. After that, enumerating solutions boils down to solving problems governed by univariate polynomials. We also provide some experiments on small-scale problems with worst-case scenarios. At the end of the paper, we propose a method for computing approximate solutions to copositive polynomial complementarity problems that may have infinitely many solutions. |
| title | Univariate representations of solutions to generic polynomial complementarity problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2410.21810 |