# Proper efficiency and cone efficiency

## TL;DR

This paper explores two general concepts of proper efficiency in vector optimization, analyzing their interrelations, properties, and conditions for existence across different spaces and specific cases like Geoffrion's efficiency.

## Contribution

It provides a comprehensive comparison of proper efficiency notions, including Henig's and Nehse and Iwanow's, and characterizes their properties using Gerstewitz functionals in topological vector spaces.

## Key findings

- Interdependencies between proper efficiency concepts are established.
- Conditions for the existence of Geoffrion's properly efficient points are proved.
- Proper efficiency notions are characterized via minimizers of convex and sublinear functionals.

## Abstract

In this report, two general concepts for proper efficiency in vector optimization are studied. Properly efficient elements can be defined as minimizers of functionals with certain monotonicity properties or as weakly efficient elements with respect to sets that contain the domination set. Interdependencies between both concepts are proved in topological vector spaces by means of Gerstewitz functionals. The investigation includes proper efficiency notions introduced by Henig and by Nehse and Iwanow. In contrary to Henig's notion, proper efficiency by Nehse and Iwanow is defined as efficiency with respect to certain convex sets which are not necessarily cones. For the finite-dimensional case, we turn to Geoffrion's proper efficiency as a special case of Henig's proper efficiency. It is characterized as efficiency with regard to subclasses of the set of polyhedral cones. Conditions for the existence of Geoffrion's properly efficient points are proved. For closed feasible point sets, Geoffrion's properly efficient point set is empty or coincides with that of Nehse and Iwanow. Properly efficient elements by Nehse and Iwanow are the minimizers of continuous convex functionals with certain monotonicity properties. Henig's proper efficiency can be described by means of minimizers of continuous sublinear functionals with certain monotonicity properties.