Epi-differentiability and optimality conditions for an extremal problem under inclusion constraints. (English) Zbl 1126.49306

Amahroq, T.; Gadhi, N.; Riahi, H.

JIPAM, J. Inequal. Pure Appl. Math. 4, No. 2, Paper No. 41, 11 p. (2003).

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.

Reviewer: Neculai Papaghiuc (Iaşi)

MSC:

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

Keywords:

optimality conditions; epi-convergence; epi-differentiability; support function