Families of differentiable scalarization functions
Let F be a family of real-valued maps on \(R^ n\), and let Y be a subset of \(R^ n\). Denote by S(Y\(| F)\) the set of all \(y^*\in Y\) such that, for some \(f\in F\), \(f(y)\geq f(y^*)\) for all y in Y. Let us say that F is a scalarization family if, for any subset Y, S(Y\(| F)\) is equal to the set of properly efficient points in Y. General conditions for F to be a scalarization family were given in a previous paper of the author [J. Optimization Theory Appl. 41, 491-502 (1983; Zbl 0518.90080)]. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of the above-cited paper which forces nondifferentiability is dropped, then S(Y\(| F)\) is dense in the set of properly efficient points.
scientific article; zbMATH DE number 4062845