T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method

TL;DR

This paper introduces a semidefinite relaxation approach to compute T-optimal designs for discriminating multi-factor polynomial regression models within constrained design spaces, extending existing methods to multi-factor cases.

Contribution

It develops a novel convex optimization framework using semidefinite programming and moment relaxations for T-optimal design in multi-factor polynomial models, including convergence guarantees.

Findings

Exact semidefinite representation for one-factor models.

Hierarchical semidefinite relaxations for multi-factor models.

Successful illustration with multiple examples.

Abstract

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality is confirmed by an equivalence theorem. The methodology is illustrated with several examples.