APPROXIMATE OPTIMAL DESIGNS FOR MULTIVARIATE POLYNOMIAL REGRESSION

We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semialgebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. T...

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Bibliographic Details
Published in:The Annals of statistics 2019-02, Vol.47 (1), p.127-155
Main Authors: De Castro, Yohann, Gamboa, Fabrice, Henrion, Didier, Hess, Roxana, Lasserre, Jean-Bernard
Format: Article
Language:English
Subjects:
Convergence
Design of experiments
Design optimization
Hierarchies
Mathematics
Multivariate analysis
Optimization and Control
Polynomials
Semidefinite programming
Statistics
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Summary:We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semialgebraic) 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.
ISSN:0090-5364
2168-8966
DOI:10.1214/18-AOS1683