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Regularity, calmness and support principle. (English) Zbl 0648.49016

The paper deals with necessary optimality conditions for the following mathematical programming problem \[ (P)\quad \min \{g(x): x\in D,\quad 0\in T(x)\}, \] where \(T: X\to Y\) is a continuous set-valued map with convex values between Banach spaces, D is a subset of X and \(g: X\to R\) is a continuous functional. A “support principle” (i.e. an analog of Lagrange multipliers for the constraint \(0\in T(x))\) is proved under a suitable regularity assumption on the constraints. It is also shown that this last condition implies the calmness of (P) in the sense of Clarke. All the results are based on a new concept of prederivative of a set- valued map.
Reviewer: G.Stefani

MSC:

49K27 Optimality conditions for problems in abstract spaces
49J52 Nonsmooth analysis
90C48 Programming in abstract spaces
26E25 Set-valued functions
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