Timetable
Tuesday, 16:30-18:00,
Freihaus of TU Wien, green area, 6th floor, seminar room 107.
Tu, 24.07.2007
Andreas H. Hamel
University Halle-Wittenberg, on leave
ORFE, Princeton University
"A duality theory for set-valued convex functions with
applications to set-valued convex risk measures"
Abstract:
Duality for extended real-valued convex functions is a well-studied,
even classical subject based on works of Fenchel, Moreau, Rockafellar,
among many others. A corresponding satisfying theory for functions
mapping into the power set of a partially ordered locally convex space
is still missing. Such a theory seems to be very desirable since it has
already been observed e.g. by Luc in 1989 that the dual of a convex
vector optimization problem 'is set-valued in nature'. Moreover, the
concept of convex set-valued risk measures has been defined recently in
financial mathematics which asks for a corresponding dual representation
theory.
We shall present a duality concept that is based on a new notion of
affine minorants for set-valued functions and show that almost every
concept (e.g. properness, sublinearity, conjugates, inf-convolution) and
result (e.g. biconjugation and Fenchel-Rockafellar duality theorems)
known in the scalar convex analysis can be established within the new
set-valued framework.
A special feature of the methodology is that proofs do not rely on the
corresponding scalar theory - as in almost every previous duality theory
for vector optimization problems. Finally, we shall show the theory at
work when applied to set-valued convex risk measures in order to give
dual representation results.