Conditions for universal reducibility of a two-stage extremization problem to a one-stage problem

Abstract

An extremization problem in the general form max x ϵ X ƒ(x) is considered. This notation is treated as the description of a parametric family of problems where the extremized function ƒ is the same but the set X is a variable parameter. Such a “mass” treatment of the extremization problem for function ƒ determines implicitly the “choice transformation” X → Y , where Y is the set of solutions, i.e., y = Arg max x ϵ X ƒ(x) , and X goes over a given family X of admissible sets. Similarly, a two-stage problem of sequential extremization of two functions, ϑ and ψ, determines the superposition of two related choice transformations. We consider the cases when the function ϑ and/or ψ may be vectorial and the extremization is understood in the Pareto sense. The very possibility of reducing a two-stage problem to a one-stage problem having the same solution set Y for every admissible X ∈ X is studied. Unlike the usual “lexicographical” extremization of two scalar functions, such a reduction is not always possible in the vectorial case. The necessary and sufficient conditions for it are stated.