Convergence rates for the moment-SoS hierarchy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916725288599552 |
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| author | Schlosser, Corbinian Tacchi, Matteo Lazarev, Alexey |
| author_facet | Schlosser, Corbinian Tacchi, Matteo Lazarev, Alexey |
| contents | We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing convergence rate analysis, initially developed for static polynomial optimization, to the more general and challenging domain of the generalized moment problem. We establish an easy-to-follow procedure for obtaining convergence rates. Our methodology is based on, firstly, a state-of-the-art degree bound for Putinar's Positivstellensatz, secondly, quantitative polynomial approximation bounds, and, thirdly, a geometric Slater condition on the infinite dimensional LP. We address a broad problem formulation that encompasses various applications, such as optimal control, volume computation, and exit location of stochastic processes. We illustrate the procedure at these three problems and, using a recent improvement on effective versions of Putinar's Positivstellensatz, we improve existing convergence rates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rates for the moment-SoS hierarchy Schlosser, Corbinian Tacchi, Matteo Lazarev, Alexey Optimization and Control We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing convergence rate analysis, initially developed for static polynomial optimization, to the more general and challenging domain of the generalized moment problem. We establish an easy-to-follow procedure for obtaining convergence rates. Our methodology is based on, firstly, a state-of-the-art degree bound for Putinar's Positivstellensatz, secondly, quantitative polynomial approximation bounds, and, thirdly, a geometric Slater condition on the infinite dimensional LP. We address a broad problem formulation that encompasses various applications, such as optimal control, volume computation, and exit location of stochastic processes. We illustrate the procedure at these three problems and, using a recent improvement on effective versions of Putinar's Positivstellensatz, we improve existing convergence rates. |
| title | Convergence rates for the moment-SoS hierarchy |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2402.00436 |