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

**Authors:** Lucy L. Gao, Julie Zhou  
**arXiv:** [1910.00745](https://arxiv.org/abs/1910.00745)  
**Date:** 2019-10-02  
**Summary page:** https://www.summarizepaper.com/en/paper/1910.00745v1/

## Key points

- Error covariance matrix is never known in practice for multi-response regression models
- Designing optimal experiments that are robust against possible misspecification of the error covariance matrix is a challenge
- Gao and Zhou propose an approach that approximates the error covariance matrix with a neighborhood of covariance matrices to define minimax D-optimal designs
- Constructing robust designs for multi-response regression models is not straightforward due to non-convex optimization problems
- Objective function for minimax D-optimal design is a difference of two convex functions, leading to the development of a flexible algorithm for computing minimax D-optimal designs that can be applied to any multi-response model with a discrete design space
- The authors derive several theoretical results for minimax D-optimal designs, including scale invariance and reflection symmetry
- Gao and Zhou's research provides valuable insights into designing optimal experiments in multi-response regression models while accounting for potential errors in the error covariance matrix.

## Summary

In the field of multi-response regression models, it is common knowledge that the error covariance matrix is never known in practice. This poses a challenge when designing optimal experiments that are robust against possible misspecification of the error covariance matrix. To address this issue, researchers Lucy L. Gao and Julie Zhou propose an approach that approximates the error covariance matrix with a neighborhood of covariance matrices. By doing so, they define minimax D-optimal designs that can withstand small deviations from an assumed error covariance matrix. Constructing robust designs for multi-response regression models is not straightforward due to the non-convex optimization problems associated with them. Even for one-response regression models, constructing robust designs analytically or numerically can be challenging. To overcome this obstacle, Gao and Zhou show that the objective function for minimax D-optimal design is a difference of two convex functions. This insight leads to the development of a flexible algorithm for computing minimax D-optimal designs that can be applied to any multi-response model with a discrete design space. The authors also derive several theoretical results for minimax D-optimal designs, including scale invariance and reflection symmetry. Overall, Gao and Zhou's research provides valuable insights into designing optimal experiments in multi-response regression models while accounting for potential errors in the error covariance matrix. Their approach offers a practical solution to an ongoing challenge faced by researchers in this field and has implications for future experimental design strategies.