Stability and scalarization in vector optimization using improvement sets (Q495731)

Assume that \(X\) and \(Y\) are normed linear spaces and \(K\) is a proper, closed, convex, pointed cone in \(Y\) with nonempty interior. A nonempty set \(E \subseteq Y\) is said to be an improvement set iff (i) \(0\not\in E\) and (ii) \(E+K=E\). When \(E\) is an improvement set, then an element \(y\in A\subset Y \) is said to be (i) an \(E\)-minimal solution of \(A\) iff \[ (A-y)\cap (-E )=\emptyset, \] (ii) a weak \(E\)-minimal solution of \(A\) iff \[ (A-y)\cap (-\mathrm{int }E )=\emptyset. \] The set of \(E\)-minimal solutions of \(A\) and the set of weak \(E\)-minimal solutions of \(A\) are denoted by \(O^E(A)\) and \(O^{\mathrm{int }E}(A ),\) respectively. Let \(f: X \rightarrow Y\) be a function and \(\emptyset\neq S \subset X\) be a closed set. The vector optimization problem (P) is given as \[ \min_ {x\in S} f (x). \] The sets of \(E\)-minimal and weak \(E\)-minimal solutions of (P) are denoted by \(O^E (f (S))\) and \(O^{\mathrm{int }E} (f (S)),\) respectively. An element \(x\in S\) is said to be an \(E\)-optimal solution of (P) iff \(f (x )\in O^E (f (S))\) and is said to be a weak \(E\)-optimal solution of (P) iff \(f (x )\in O^{\mathrm{int } E} (f (S)), \) respectively. In this article, the authors established the lower and upper set-convergences of \(E\)-optimal and weak \(E\)-optimal solution sets of perturbed nonconvex vector problems in the sense of Painlevé-Kuratowski. In fact, they investigated the stability of \(E\)-optimal solution sets of a family of perturbed vector problems in terms of Painlevé-Kuratowski set-convergence. Furthermore, they studied these convergences through the set-convergences of the approximate solution sets of certain scalarized problems by using a nonlinear scalarization function defined in terms of an improvement set.