Analytic Solutions for D-optimal Factorial Designs under Generalized Linear Models

We develop two analytic approaches to solve D-optimal approximate designs under generalized linear models. In particular, we provide analytic D-optimal designs under $2^2$ main-effects model considered by Yang, Mandal and Majumdar (2012). As an application of the analytic solution developed, we provide a necessary and sufficient condition under which a D-optimal design with two quantitative factors could be constructed on the boundary points only. It bridges the gap between D-optimal factorial designs and D-optimal designs with continuous factors.