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Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. (English) Zbl 0791.49018
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:
49J52Nonsmooth analysis (other weak concepts of optimality)
46A30Open mapping and closed graph theorems; completeness
26E25Set-valued real functions
58C06Set-valued and function-space valued mappings on manifolds
58C20Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
49K40Sensitivity, stability, well-posedness of optimal solutions
90C48Programming in abstract spaces