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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.00838 |
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| _version_ | 1866909196448956416 |
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| author | Hieu, Vu Trung Takeda, Akiko |
| author_facet | Hieu, Vu Trung Takeda, Akiko |
| contents | In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. By using a technique in computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e. the constraint set is $\R^n$, the coordinates of all local minimizers can be represented by the values of $n$ univariate polynomials at real roots of a system including a univariate polynomial equation and a univariate polynomial matrix inequality. We also develop the technique for constrained problems having equality/inequality constraints. Based on the above technique, we design symbolic algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper, we propose a perturbation technique to compute approximately a global minimizer provided that the constraint set is compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00838 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computing local minimizers in polynomial optimization under genericity conditions Hieu, Vu Trung Takeda, Akiko Optimization and Control In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. By using a technique in computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e. the constraint set is $\R^n$, the coordinates of all local minimizers can be represented by the values of $n$ univariate polynomials at real roots of a system including a univariate polynomial equation and a univariate polynomial matrix inequality. We also develop the technique for constrained problems having equality/inequality constraints. Based on the above technique, we design symbolic algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper, we propose a perturbation technique to compute approximately a global minimizer provided that the constraint set is compact. |
| title | Computing local minimizers in polynomial optimization under genericity conditions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2311.00838 |