# Approximate Optimal Designs for Multivariate Polynomial Regression

## TL;DR

This paper presents a novel semidefinite programming approach using the moment-sum-of-squares hierarchy to compute approximate optimal designs for multivariate polynomial regression on semi-algebraic spaces, ensuring convergence and providing dual certificates.

## Contribution

It introduces a new method leveraging semidefinite programming and duality theory to compute and certify approximate optimal designs with proven convergence properties.

## Key findings

- Hierarchy converges to the optimal design as order increases
- Dual certificates guarantee finite convergence
- Revisits the experimental design equivalence theorem

## 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.