Directional derivatives and subdifferentials of set-valued convex functions

Andreas H. Hamel; Carola Schrage

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arXiv · 1207.5295

Directional derivatives and subdifferentials of set-valued convex functions

Andreas H. Hamel·Carola Schrage

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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.

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Andreas H. Hamel, Carola Schrage. 2012-07-23. Directional derivatives and subdifferentials of set-valued convex functions. https://arxiv.org/abs/1207.5295

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