Linear separation for G-semidifferentiable problems

This paper aims to study, in the image space, a certain approximation of a constrained extremum problem: the approximation is obtained by substituting the functions involved with their G-derivatives. It is shown that, in the hypothesis of G-differentiability, the linear separation between a given convex set and the image of the approximated problem is equivalent to the semistationarity of the Lagrangian function associated to the original problem. Similar results are obtained considering approximations involving the Dini-Hadamard derivatives.