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Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube

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Název: Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube
Autoři: Gribling, Sander, de Klerk, Etienne, Vera, Juan
Rok vydání: 2025
Sbírka: ArXiv.org (Cornell University Library)
Témata: Optimization and Control
Popis: We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the minimum value $f^*$, where $r$ refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter $r$. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order $1/r$. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order $1/r^2$ for our approach. ; 24 pages, 3 figures
Druh dokumentu: text
Jazyk: unknown
Relation: http://arxiv.org/abs/2505.00544
Dostupnost: http://arxiv.org/abs/2505.00544
Přístupové číslo: edsbas.AA21A4E4
Databáze: BASE
Popis
Abstrakt:We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the minimum value $f^*$, where $r$ refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter $r$. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order $1/r$. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order $1/r^2$ for our approach. ; 24 pages, 3 figures