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Continuity of set-valued maps revisited in the light of tame geometry

Aris Daniilidis(arisd***at***mat.uab.es)
C.H. Jeffrey Pang(cp229***at***cornell.edu)

Abstract: Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result –depending on stratification techniques– holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.

Keywords: Set-valued map, (strict, outer, inner) continuity, Aubin property, semialgebraic, piecewise polyhedral, tame optimization.

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: preprint, CRM, 2009

Download: [PDF]

Entry Submitted: 05/04/2009
Entry Accepted: 05/04/2009
Entry Last Modified: 05/04/2009

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