Convergence rates of S.O.S hierarchies for polynomial semidefinite programs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917021852106752 |
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| author | Tran, Hoang Anh Toh, Kim-Chuan |
| author_facet | Tran, Hoang Anh Toh, Kim-Chuan |
| contents | We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the Łojasiewicz inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rates of S.O.S hierarchies for polynomial semidefinite programs Tran, Hoang Anh Toh, Kim-Chuan Optimization and Control 90C22, 90C26, 41A10, 41A50 We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the Łojasiewicz inequality. |
| title | Convergence rates of S.O.S hierarchies for polynomial semidefinite programs |
| topic | Optimization and Control 90C22, 90C26, 41A10, 41A50 |
| url | https://arxiv.org/abs/2406.12013 |