Computing optimal designs for regression models via convex programming




Zhou, Wenjie

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Optimal design problems aim at selecting design points optimally with respect to certain statistical criteria. The research of this thesis focuses on optimal design problems with respect to A-, D- and E-optimal criteria, which minimize the trace, determinant and largest eigenvalue of the information matrix, respectively. Semide nite programming (SDP) is concerned with optimizing a linear objective function subject to a linear matrix being positive semide nite. Two powerful MATLAB add-ons, SeDuMi and CVX, have been developed to solve SDP problems e ciently. In this paper, we show in detail how to formulate A- and E-optimal design problems as SDP problems and solve them by SeDuMi and CVX. This technique can be used to construct approximate A-optimal and E-optimal designs for all linear and non-linear models with discrete design spaces. The results can also provide guidance to nd optimal designs on continuous design spaces. For one variable polynomial regression models, we solve the A- and E- optimal designs on the continuous design space by using a two-stage procedure. In the rst stage we nd the optimal moments by casting it as an SDP problem and in the second stage we extract the optimal designs from the optimal moments obtained from the rst stage. Unlike E- and A-optimal design problems, the objective function of D-optimal design problem is nonlinear. So D-optimal design problems cannot be reformulated as an SDP. However, it can be cast as a convex problem and solved by an interior point method. In this thesis we give details on how to use the interior point method to solve D-optimal design problems. Finally several numerical examples for A-, D-, and E-optimal designs along with the MATLAB codes are presented.



A-optimality, approximate design, convex programming, CVX, D-optimality, E-optimality, interior point method, SeDuMi, semidefinite programming