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Zbl 1225.90115
Durea, M.; Strugariu, R.
Necessary optimality conditions for weak sharp minima in set-valued optimization. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 7, 2148-2157 (2010). ISSN 0362-546X

Summary: The aim of the present paper is to get necessary optimality conditions for a general kind of sharp efficiency for set-valued mappings in infinite dimensional framework. The efficiency is taken with respect to a closed convex cone and as the basis of our conditions we use the Mordukhovich generalized differentiation. We have divided our work into two main parts concerning, on the one hand, the case of a solid ordering cone and, on the other hand, the general case without additional assumptions on the cone. In both situations, we derive some scalarization procedures in order to get the main results in terms of the Mordukhovich coderivative, but in the general case we also carryout a reduction of the sharp efficiency to the classical Pareto efficiency which, in addition with a new calculus rule for Fréchet coderivative of a difference between two maps, allows us to obtain some results in Fréchet form.

MSC 2000:: *90C29 Multi-objective programming, etc.
90C26 Nonconvex programming
49J52 Nonsmooth analysis (other weak concepts of optimality)

Keywords: set-valued optimization; sharp efficiency; Mordukhovich generalized differentiation; Fréchet normal cone

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