ABSTRACT
In this chapter we discuss optimal designs under constraints. The idea of incorporating constraints (costs, or penalties) in optimal design goes back to the seminal paper by Elfving (1952) [114] on the geometrical construction of coptimal designs: at the very end of the paper, Elfving introduced a constraint on the total costs instead of the constraint on the number of observations. In the notation of this chapter, the former corresponds to (4.6) while the latter corresponds, as usual, to (4.5). Elfving introduced the optimal design problem with the single constraint, and as we show in Section 4.1, for normalized designs that problem can be rather easily reduced to the standard setting. The problem with multiple constraints is much more challenging; see a survey paper by Cook and Fedorov (1995) [79] for the relevant discussion. For other references, see Cook and Wong (1994) [80], Mentre´ et al. (1997) [277], Fedorov et al. (2002) [134].
