[FAM-news] reminder, this week's seminars

Sandra Trenovatz sandra at fam.tuwien.ac.at
Mon Jul 23 18:17:36 CEST 2007


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"


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 

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.

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