A duality theory for set-valued functions: Fenchel conjugationtheory

In this thesis, Hamel's article was studied \cite{hame3}. Fenchel conjugate theory is taken as a point of view. A generalized condition of being convex for set-valued maps is recalled. Conditions of being proper and closed for such a set-valued mapping are also studied. The concepts of convex analysis for set-valued functions are obtained similar to do scalar case. It is given that a set-valued closed convex function can be interpreted as the point supremum of the continuous affine minorants of this functions. In this way, the suitability of a function to the Fenchel Conjugate and Moreau-Fenchel Theorem was evaluated. A preordered, discrete local convex space proved to be the point supremum of set-valued affine minorants of a unique closed convex function having values in the power set. A new Legendre-Fenchel concept for set-valued functions is defined and the Moreau-Fenchel Theorem is proved. Examples and applications were given. Among them, a dual representation theorem for set-valued convex risk measures was also given.