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Zbl 0791.49018
Mordukhovich, Boris
Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. (English)
[J] Trans. Am. Math. Soc. 340, No.1, 1-35 (1993). ISSN 0002-9947; ISSN 1088-6850/e

Summary: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings. The results obtained provide complete characterizations of the properties under consideration in a general setting of closed-graph multifunctions in finite dimensions. They ensure new information even in the classical cases of smooth single- valued mappings as well as multifunctions with convex graphs.
MSC 2000:
*49J52 Nonsmooth analysis (other weak concepts of optimality)
46A30 Open mappings and closed graph theorems, etc.
26E25 Set-valued real functions
58C06 Set-valued mappings etc. on manifolds
58C20 Generalized differentiation theory on manifolds
49K40 Sensitivity of optimal solutions in the presence of perturbations
90C48 Programming in abstract spaces

Keywords: multifunctions; set-valued mappings; inverse mapping; robust generalized derivatives; nonsmooth mappings; closed-graph multifunctions

Cited in: Zbl 1176.49032 Zbl 1173.49023 Zbl 1212.49022 Zbl 1155.49008 Zbl 1137.47038

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