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Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel
Consider the problem of minimizing a polynomial over a compact semialgebraic set . 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 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 at a rate in , matching recently obtained convergence rates for the hypersphere and hypercube . 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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