Isolated and proper efficiencies in semi-infinite vector optimization problems (Q467444)

The paper addresses semi-infinite vector optimization problems of the form \(\min_{\mathbb{R}_+^m} \{f(x): x \in C\}\), where \(C\) is defined by a possibly infinite set of inequality constraints. It is concerned with local efficient, local strongly isolated, and local positively properly efficient solutions of such problems. The authors prove a sufficient condition for local strongly isolated solutions under a limiting constraint qualification. If the vectors of the objective and constraint functions are locally convex at a point that satisfies the condition, they show that the condition is sufficient. Furthermore, they prove a necessary condition for local positively properly efficient solutions under the limiting constraint qualification. This condition is shown to be sufficient under generalized convexity. Finally, a Wolfe-type dual problem is defined for which weak and strong duality results are proved.