Darinka Dentcheva
Regular Castaing Representations of Multifunctions with Applications to Stochastic Programming
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections, See Also {26E25, 54C60, 54C65, 90A14}

90C15 Stochastic programming

53C65 Integral geometry, See also {52A22, 60D05}; differential forms, currents, etc. See mainly{58Axx}

62G20 Asymptotic properties

Abstract: We consider set-valued mappings defined on a topological space with convex closed images in Rⁿ. The measurability of a multifunction is characterized by the existence of a Castaing representation for it: a countable set of measurable selections that pointwise fill up the graph of the multifunction. Our aim is to construct a Castaing representation which inherits the regularity properties of the multifunction. The construction uses Steiner points. A notion of a generalized Steiner point is introduced. A Castaing representation called regular is defined by using generalized Steiner selections. All selections are measurable, continuous, resp. Hölder-continuous, or directionally differentiable, if the multifunction has the corresponding properties. The results are applied to various multifunctions arising in stochastic programming. In particular, statements about the asymptotic behavior of measurable selections of solution sets via the delta-method are obtained.
Keywords: Steiner center, selections, Cataing representation, stochastic programs, random sets, delta-theorems