E-optimal designs for second order response surface models

Abstract

E-optimal experimental designs for a second order response surface model with k>=1 predictors are investigated. If the design space is the k-dimensional unit cube, Galil and Kiefer (1977a) determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper we prove this claim and show that these designs are in fact E-optimal in the class of all approximate designs. Morever, if the design space is the unit ball, E-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem. The main difficulty in the construction of E-optimal designs for the second order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the optimal information matrix" is larger than one (in contrast to the case k = 1) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise form a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding E-optimal design problems.