Cone-valued maps in optimization

Abstract

Cone-valued maps are special set-valued maps where the image sets are cones. Such maps play an important role in optimization, for instance in optimality conditions or in the context of Bishop–Phelps cones. In vector optimization with variable ordering structures, they have recently attracted even more interest. We show that classical concepts for set-valued maps as cone-convexity or monotonicity are not appropriate for characterizing cone-valued maps. For instance, every convex or monotone cone-valued map is a constant map. Similar results hold for cone-convexity, sublinearity, upper semicontinuity or the local Lipschitz property. Therefore, we also propose new concepts for cone-valued maps.

Keywords:

• cone-valued map
• set-valued map
• convex
• linear
• continuous
• monotone

AMS Subject Classifications::

• 90C30
• 90C29

Acknowledgements

The author is grateful to Truong Xuan Duc Ha and Marco Pruckner for valuable discussions and to the anonymous referees for helpful remarks on the first version of this article.