A Fenchel–Rockafellar duality theorem for set-valued optimization

Abstract

A duality theorem of the Fenchel–Rockafellar type for set-valued optimization problems is presented along with a result for the conjugate of the sum of two set-valued functions and a chain rule. The underlying solution concepts rely on order complete lattices of sets defined via set relations. Set-valued replacements for linear operators are used as dual variables, for example in order to define Fenchel conjugates for set-valued functions. An application to mathematical finance is given.