Sums of squares certificates for polynomial moment inequalities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911873137377280 |
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| author | Klep, Igor Magron, Victor Volčič, Jurij |
| author_facet | Klep, Igor Magron, Victor Volčič, Jurij |
| contents | This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. A positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_05761 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sums of squares certificates for polynomial moment inequalities Klep, Igor Magron, Victor Volčič, Jurij Functional Analysis Optimization and Control Quantum Physics 13J30, 44A60, 60E15, 90C22, 46G12, 47L60, 81-08 This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. A positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled. |
| title | Sums of squares certificates for polynomial moment inequalities |
| topic | Functional Analysis Optimization and Control Quantum Physics 13J30, 44A60, 60E15, 90C22, 46G12, 47L60, 81-08 |
| url | https://arxiv.org/abs/2306.05761 |