Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel
Abstract
Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel--Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.
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© 2022, Society for Industrial and Applied Mathematics.
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Submitted: 9 November 2021
Accepted: 3 May 2022
Published online: 3 November 2022
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H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 764759
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