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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.

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