Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities
The paper proves results on the stability of weak efficient, efficient, and Henig proper efficient sets of constrained vector optimization problems \(\min \{f(x): x \in S\}\) with quasiconvex objective functions. The authors first recall the definitions of quasiconvexity, efficiency, and of the Kuratowski-Painlevé set convergence of sets. They consider a sequence \((P_n)\) of perturbed vector optimization problems \(\min\{f_n(x): x \in S_n\}\) such that \(S_n\) Kuratowski-Painlevé converges to \(S\) and \(f_n\) \(\Gamma\)-converges to \(f\). The results show Kuratowski-Painlevé convergence of weak minimal and minimal sets under quasi-convexity of \(f_n\) and strict quasi-convexity of \(f\). Examples illustrate that the results do not hold without the strict quasiconvexity assumption. Next, the convergence of the sets of (weak) efficient solutions under the same conditions is proved. These results are then extended to the sets of Henig proper minimal sets and Henig proper efficient sets. Finally, results on the Kuratowski-Painlevé convergence of solution sets of scalarized problems with (proper, strict) order preserving functions to (Henig proper, weak) efficient solution sets of the vector optimization problem are proved.
scientific article; zbMATH DE number 6136262