Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

Lucas Slot

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arXiv · 2111.04610

Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

Lucas Slot

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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 Schm\"udgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schm\"udgen-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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Lucas Slot. 2021-11-08. Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel. https://arxiv.org/abs/2111.04610

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