Abstract : We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. This article shows that the hierarchy converges to the approximate optimal design as the order of the hierarchy increases. Furthermore, we provide a dual certificate ensuring finite convergence of the hierarchy and showing that the approximate optimal design can be computed numerically with our method. As a byproduct, we revisit the equivalence theorem of the experimental design theory: it is linked to the Christoffel polynomial and it characterizes finite convergence of the moment-sum-of-square hierarchies.
https://hal.laas.fr/hal-01483490
Contributeur : Didier Henrion
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Soumis le : mardi 17 octobre 2017 - 10:41:26
Dernière modification le : vendredi 14 septembre 2018 - 09:16:06
Document(s) archivé(s) le : jeudi 18 janvier 2018 - 12:42:10
Yohann De Castro, Fabrice Gamboa, Didier Henrion, Roxana Hess, J.-B Lasserre. Approximate Optimal Designs for Multivariate Polynomial Regression. Rapport LAAS n° 17044. To appear at Annals of Satistics. 2017. 〈hal-01483490v2〉
