Sum-of-squares hierarchies for polynomial > optimization and the Christoffel-Darboux kernel
Slot,Lucas
Slot,Lucas
Abstract
Consider the problem of minimizing a polynomial f over a compact semialgebraic set X⊆Rn. 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 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/r2), 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.
Description
Publisher Copyright: 2022 Society for Industrial and Applied Mathematics.
Date
2022-12
Journal Title
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Volume Title
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Research Projects
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Keywords
Christoffel-Darboux kernel, Positivstellensatz, polynomial kernel method, polynomial optimization, sum-of-squares hierarchy
Citation
Slot, L 2022, 'Sum-of-squares hierarchies for polynomial > optimization and the Christoffel-Darboux kernel', SIAM Journal on Optimization, vol. 32, no. 4, pp. 2612-2635. https://doi.org/10.1137/21M1458338
License
info:eu-repo/semantics/closedAccess