Sum-of-squares relaxations for polynomial min–max problems over simple sets
We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, t...
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| Vydáno v: | Mathematical programming Ročník 209; číslo 1; s. 475 - 501 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer Berlin Heidelberg
01.01.2025
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar’s Positivstellensatz. |
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| AbstractList | We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar’s Positivstellensatz. |
| Author | Bach, Francis |
| Author_xml | – sequence: 1 givenname: Francis surname: Bach fullname: Bach, Francis email: francis.bach@inria.fr organization: Inria, Ecole Normale Supérieure, PSL Research University |
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| Cites_doi | 10.1109/ACC.2001.945730 10.2140/pjm.1958.8.171 10.1109/CDC.2004.1428957 10.1007/10997703_15 10.1007/s11590-022-01922-5 10.1137/S1052623494267127 10.1137/100814147 10.1017/CBO9780511804441 10.1007/s10107-020-01537-7 10.1007/s10107-005-0684-2 10.1142/q0252 10.1007/s10208-021-09526-8 10.1007/s10898-010-9628-3 10.1515/9781400831050 10.1137/0915077 10.1137/S1052623400366802 10.1512/iumj.1993.42.42045 10.1007/s10107-003-0387-5 10.1007/978-3-030-42760-3 10.1137/22M1540818 10.1007/BF00968419 10.1142/9134 10.1007/978-3-319-64546-9 10.1007/s10107-010-0353-y 10.1137/0806020 |
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| Keywords | 11E25 Semidefinite programming Sum-of-squares Polynomial optimization 90C22 Min–max problems |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical |
| Title | Sum-of-squares relaxations for polynomial min–max problems over simple sets |
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