Convergence rate for linear minimizer-estimators in the moment-sum-of-squares hierarchy
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913706751819776 |
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| author | Schlosser, Corbinian |
| author_facet | Schlosser, Corbinian |
| contents | Effective Positivstellensätze provide convergence rates for the moment-sum-of-squares (SoS) hierarchy for polynomial optimization (POP). In this paper, we add a qualitative property to the recent advances in those effective Positivstellensätze. We consider optimal solutions to the moment relaxations in the moment-SoS hierarchy and investigate the measures they converge to. It has been established that those limit measures are the probability measures on the set of optimal points of the underlying POP. We complement this result by showing that these measures are approached with a convergence rate that transfers from the (recent) effective Positivstellensätze. As a special case, this covers estimating the minimizer of the underlying POP via linear pseudo-moments. Finally, we analyze the same situation for another SoS hierarchy - the upper bound hierarchy - and show how convexity can be leveraged. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence rate for linear minimizer-estimators in the moment-sum-of-squares hierarchy Schlosser, Corbinian Optimization and Control Effective Positivstellensätze provide convergence rates for the moment-sum-of-squares (SoS) hierarchy for polynomial optimization (POP). In this paper, we add a qualitative property to the recent advances in those effective Positivstellensätze. We consider optimal solutions to the moment relaxations in the moment-SoS hierarchy and investigate the measures they converge to. It has been established that those limit measures are the probability measures on the set of optimal points of the underlying POP. We complement this result by showing that these measures are approached with a convergence rate that transfers from the (recent) effective Positivstellensätze. As a special case, this covers estimating the minimizer of the underlying POP via linear pseudo-moments. Finally, we analyze the same situation for another SoS hierarchy - the upper bound hierarchy - and show how convexity can be leveraged. |
| title | Convergence rate for linear minimizer-estimators in the moment-sum-of-squares hierarchy |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2502.18370 |