Directional derivatives and subdifferentials of set-valued convex functions
Optimization and Control
2012-07-24 v1
Abstract
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex optimization problems with a set-valued objective. As a major tool, a residuation operation is used which acts in a space of closed convex, but not necessarily bounded subsets of a topological linear space. The residuation serves as a substitute for the inverse addition and is intimately related to the Minkowski or geometric difference of convex sets. The results, when specialized, even extend those for extended real-valued convex functions since the improper case is included.
Keywords
Cite
@article{arxiv.1207.5295,
title = {Directional derivatives and subdifferentials of set-valued convex functions},
author = {Andreas H. Hamel and Carola Schrage},
journal= {arXiv preprint arXiv:1207.5295},
year = {2012}
}