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Epi-differentiability and optimality conditions for an extremal problem under inclusion constraints. (English) Zbl 1126.49306
The authors establish first-order optimality conditions for the problem of minimizing a function $$f$$ on the solution set of an inclusion $$0\in F(x)$$, where $$f$$ and the support function of a set-valued mapping $$F$$ are epi-differentiable at $$\overline{x}$$. The paper is organized as follows: Section 2 contains basic definitions and preliminaries that are widely used in the sequel. In Section 3 the authors study the epi-differentiability of the support function of $$F$$ defined by $$C_F(y^*,c):=\text{ inf}_{y\in F(x)}\langle y^*,y\rangle$$ for every $$y^*\in Y^*$$. Section 4 is devoted to the optimality conditions and also for an application in mathematical programming problems.

##### MSC:
 49J52 Nonsmooth analysis 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds 90C46 Optimality conditions and duality in mathematical programming
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