Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916359700480000 |
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| author | Slot, Lucas Wiedmer, Manuel |
| author_facet | Slot, Lucas Wiedmer, Manuel |
| contents | Let $\mathbf{X} \subseteq \mathbb{R}^n$ be a closed set, and consider the problem of computing the minimum $f_{\min}$ of a polynomial $f$ on $\mathbf{X}$. Given a measure $μ$ supported on $\mathbf{X}$, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on $f_{\min}$, each of which may be computed by solving a semidefinite program. When $\mathbf{X}$ is compact, these bounds converge to $f_{\min}$ under minor assumptions on $μ$. Later, Lasserre (Math. Program. 190, 2020) introduces a related, but far more economical sequence of upper bounds which rely on the push-forward measure of $μ$ by $f$. While these new bounds are weaker a priori, they actually achieve similar asymptotic convergence rates on compact sets. In this work, we show that no such free lunch exists in the non-compact setting. While convergence of the standard bounds to $f_{\min}$ is guaranteed when $\mathbf{X} = \mathbb{R}^n$ and $μ$ is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to $f_{\min}$ in that setting already for polynomials of degree $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures Slot, Lucas Wiedmer, Manuel Optimization and Control 90C22, 90C23, 90C26, Let $\mathbf{X} \subseteq \mathbb{R}^n$ be a closed set, and consider the problem of computing the minimum $f_{\min}$ of a polynomial $f$ on $\mathbf{X}$. Given a measure $μ$ supported on $\mathbf{X}$, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on $f_{\min}$, each of which may be computed by solving a semidefinite program. When $\mathbf{X}$ is compact, these bounds converge to $f_{\min}$ under minor assumptions on $μ$. Later, Lasserre (Math. Program. 190, 2020) introduces a related, but far more economical sequence of upper bounds which rely on the push-forward measure of $μ$ by $f$. While these new bounds are weaker a priori, they actually achieve similar asymptotic convergence rates on compact sets. In this work, we show that no such free lunch exists in the non-compact setting. While convergence of the standard bounds to $f_{\min}$ is guaranteed when $\mathbf{X} = \mathbb{R}^n$ and $μ$ is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to $f_{\min}$ in that setting already for polynomials of degree $6$. |
| title | Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures |
| topic | Optimization and Control 90C22, 90C23, 90C26, |
| url | https://arxiv.org/abs/2404.09710 |