Conditions for universal reducibility of a two-stage extremization problem to a one-stage problem
An extremization problem in the general form \(\max_{x\in X}f(x)\) is considered. This notation is treated as the description of a parametric family of problems where the extremized function f is the same but the set X is a variable parameter. Such a ''mass'' treatment of the extremization problem for function f determines implicitly the ''choice transformation'' \(X\to Y\), where Y is the set of solutions, i.e., \(Y=Arg \max_{x\in X}f(x)\), and X varies over a given family \({\mathcal X}\) of admissible sets. Similarly, a two-stage problem of sequential extremization of two functions, \(\phi\) and \(\psi\), determines the superposition of two related choice transformations. We consider the cases when the function \(\phi\) and/or \(\psi\) 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 \(X\in {\mathcal X}\) is studied. Unlike the usual ''lexicographical'' extremization of two scalar functions, such a reduction is not always possible in the vectorial case. Necessary and sufficient conditions for it are stated.
scientific article; zbMATH DE number 4001884