Characterization of the equality of weak efficiency and efficiency on convex free disposal hulls
In solving a multi-objective optimization problem by scalarization techniques, solutions to a scalarized problem are, in general, weakly efficient rather than efficient to the original problem. Thus, it is crucial to understand what problem ensures that all weakly efficient solutions are efficient. In this paper, we give a characterization of the equality of the weakly efficient set and the efficient set, provided that the free disposal hull of the domain is convex. Using this characterization, we see that the set of weakly efficient solutions is equal to the set of efficient solutions (and also the set of strictly efficient solutions) in strongly convex problems. As practical applications, we consider the structure of the set of efficient solutions to a location problem under Mahalanobis distances and a multi-objective reformulation of the elastic net.