Title: Proper efficiency and cone efficiency,
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.