Proximal proper efficiency in set-valued optimization

In this paper, we introduce the concept of cone semilocal convex and cone semilocal convexlike set-valued maps and obtain characterization of these maps in terms of locally star-shaped sets. We derive an alternative theorem involving cone semilocal convexlike set-valued maps under the assumption of closedness of the translation of the image set of the map by the cone under consideration. We introduce proximal proper efficiency for a set-valued optimization problem in finite-dimensional spaces and obtain certain scalarization and Lagrange multiplier theorems. In the end, we consider a Lagrange form of dual and establish weak and strong duality theorems.