Directional derivatives, subdifferentials and optimality conditions for set-valued convex functions
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SubjectDirectional derivatives; Subdifferential; Set-valued function; optimality condition; max-formula; residuation
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the "max formula" is proven. The new concepts are used to provide optimality conditions for convex minimization problems with a set-valued objective. As a major tool, residuation operations are used which act on spaces 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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