Scalar representation and conjugation of set-valued functions

Abstract

We consider functions with values in the power set of a pre-ordered, separated locally convex space with closed convex images. To each such function, a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre–Fenchel conjugate for set-valued functions is introduced and identified with the conjugates of the scalarizations. To the set-valued conjugate, a full calculus is provided, including a biconjugation theorem, a chain rule and weak and strong duality results of the Fenchel–Rockafellar type.