Characterization of the equality of weak efficiency and efficiency on convex free disposal hulls

TL;DR

This paper characterizes when weakly efficient solutions are also efficient in multi-objective optimization with convex free disposal hulls, providing theoretical insights and practical applications like multi-objective LASSO.

Contribution

It offers a new characterization of the equality between weakly efficient and efficient solutions under convex free disposal hulls, with applications to optimization problems.

Findings

Weakly efficient solutions coincide with efficient solutions under certain conditions.

The characterization applies to multi-objective LASSO, ensuring all solutions are efficient.

Mathematical applications derived from the characterization enhance understanding of efficiency sets.

Abstract

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. By using this characterization, we obtain various mathematical applications. As a practical application, we show that all weakly efficient solutions to a multi-objective LASSO with mild modification are efficient.