An optimization problem with applications to optimal design theory. (English) Zbl 0623.62069

Cheng, Ching-Shui

Ann. Stat. 15, 712-723 (1987).

This paper deals with the problem of minimizing \(\sum^{n}_{i=1}f(x_ i)\) subject to the following constraints: \[ \sum^{n}_{i=1}x_ i=A,\quad \sum^{n}_{i=1}g(x_ i)=B,\quad and\quad x_ i\geq 0,\quad i=1,2,...,n \] where \(A>0\), B are fixed constants, and f, g are real- valued functions. The above problem arises, besides other areas, in the theory of optimal experimental design where \(x_ 1,x_ 2,...,x_ n\) are the eigenvalues of an (n\(\times n)\) symmetric nonnegative definite matrix (called the information matrix) \(C_ d\) corresponding to a design d in the class D of all the competing designs.
The results obtained here are then used to show that certain designs which are optimal with respect to two criteria are also optimal with respect to many others.

Reviewer: D.V.Chopra

MSC:

62K05	Optimal statistical designs
90C30	Nonlinear programming

Keywords:

optimization problem; minimization problem; A-optimality; D-optimality; E-optimality; phi-optimality; information matrix