Using SeDuMi to find various optimal designs for regression models

We introduce a powerful and yet seldom used numerical approach in statistics for solving a broad class of optimization problems where the search space is discretized. This optimization tool is widely used in engineering for solving semidefinite programming (SDP) problems and is called self-dual minimization (SeDuMi). We focus on optimal design problems and demonstrate how to formulate A-, As\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_s$$\end{document}-, c-, I-, and L-optimal design problems as SDP problems and show how they can be effectively solved by SeDuMi in MATLAB. We also show the numerical approach is flexible by applying it to further find optimal designs based on the weighted least squares estimator or when there are constraints on the weight distribution of the sought optimal design. For approximate designs, the optimality of the SDP-generated designs can be verified using the Kiefer–Wolfowitz equivalence theorem. SDP also finds optimal designs for nonlinear regression models commonly used in social and biomedical research. Several examples are presented for linear and nonlinear models.