Variational analysis for set-valued functions
Time and place
1 PM on Wednesday, March 10th, 2010; NAC 1511
Prof. Andreas Hamel (Princeton University)
Abstract
A function with values in the power set of a vector space is usually understood as a correspondence (Klein/Thompson 1984), or a multi-valued or set-valued mapping (Aubin/Frankowska 1990). This point of view makes it difficult to extend many useful concepts from real-valued to set-valued functions.
Most importantly, persuasive definitions for the infimum and supremum of a set- valued function were lacking until very recently. For this reason, a satisfactory duality theory for vector optimization problems could not yet be formulated, although it is known for a long time that the dual of a vector optimization problem is a set-valued one (Luc 1989).
In this talk, we will show that these difficulties (and many more) can be overcome by understanding a set-valued function as a function into a well-defined subset of the power set of a (pre-ordered) vector space. These image spaces of set-valued functions turn out to carry interesting algebraic and order structures. Using these structures, one can define infima/suprema, (directional) derivatives, subdifferentials, Legendre-Fenchel transforms and construct dual problems of optimization problems with a set-valued objective. Surprisingly, within this framework (almost) every result from the scalar theory has a set-valued counterpart - this definitely is not possible for vector-valued functions. Exemplarily, set-valued versions of the Fenchel-Moreau theorem and the Fenchel-Rockafellar duality theorem from Convex Analysis are shown. Unexpectedly, but highly satisfactory, an appropriate mathematical model for finan- cial markets with transactions costs can only be established using set-valued functions. Risk measures, super-hedging and indifference prices in such markets can be defined in accordance with practical needs using the above described tools. This may serve as a cutting-edge application of the presented theory.
