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Necessary optimality conditions in terms of convexificators in Lipschitz optimization. (English) Zbl 1143.90035
Recently, the idea of convexificators has been used to extend and strengthen various results in nonsmooth analysis and optimization. Convexificators, generating upper convex and lower concave approximations to a function at a point, can be viewed as weaker versions of the notation of subdifferentials that so much pervades the study of nonsmooth analysis.
The present paper considers constraint qualifications and Kuhn-Tucker type necessary optimality conditions for optimization problems with arbitrary set constraints or inequality constraints, where the objective and the constraint functions are real-valued and locally Lipschitz continuous. The qualifications and the necessary optimality conditions are stated in terms of upper or lower convexificators. First, the minimization problem subject to an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.

MSC:
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
90C47 Minimax problems in mathematical programming
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