# D-optimal design for multivariate polynomial regression via the Christoffel function and semidefinite relaxations

## TL;DR

This paper introduces a novel method for designing D-optimal experiments in multivariate polynomial regression using semidefinite programming and Christoffel functions, enabling efficient numerical solutions and geometric interpretation.

## Contribution

It develops a new approach combining moment-sum-of-squares hierarchy and Christoffel polynomial for optimal experimental design in polynomial regression.

## Key findings

- Effective numerical approximation of D-optimal designs.
- Utilization of semidefinite programming duality for geometric insights.
- Applicable to compact semi-algebraic design spaces.

## Abstract

We present a new approach to the design of D-optimal experiments with multivariate polynomial regressions on compact semi-algebraic design spaces. We apply the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically and approximately the optimal design problem. The geometry of the design is recovered with semidefinite programming duality theory and the Christoffel polynomial.