Title: Approximate Optimal Designs for Multivariate Polynomial Regression,
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.