Minimax D-optimal designs for multivariate regression models with multi-factors

TL;DR

This paper develops a novel approach for constructing minimax D-optimal designs in multivariate regression models that are robust to errors in the covariance matrix, using a difference-of-convex-functions optimization method.

Contribution

It introduces a new algorithm based on difference-of-convex-functions optimization for robust design construction in multi-response regression models.

Findings

The proposed algorithm effectively computes minimax D-optimal designs.

Theoretical properties like scale invariance and reflection symmetry are established.

The method applies to any multi-response model with a discrete design space.

Abstract

In multi-response regression models, the error covariance matrix is never known in practice. Thus, there is a need for optimal designs which are robust against possible misspecification of the error covariance matrix. In this paper, we approximate the error covariance matrix with a neighbourhood of covariance matrices, in order to define minimax D-optimal designs which are robust against small departures from an assumed error covariance matrix. It is well known that the optimization problems associated with robust designs are non-convex, which makes it challenging to construct robust designs analytically or numerically, even for one-response regression models. We show that the objective function for the minimax D-optimal design is a difference of two convex functions. This leads us to develop a flexible algorithm for computing minimax D-optimal designs, which can be applied to any multi-response model with a discrete design space. We also derive several theoretical results for minimax D-optimal designs, including scale invariance and reflection symmetry.