Approximate optimal designs for multivariate polynomial regression
Yohann De Castro, Fabrice Gamboa, Didier Henrion, Roxana Hess, Jean-Bernard Lasserre
The Annals of Statistics · 2019 · Volume 47 · Issue 1
Abstract
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.
Keywords
15A15, 41A10, 49M29, 62K05, 90C25, 90C90, Christoffel polynomial, equivalence theorem, Experimental design, linear model, semidefinite programming
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