Characterizing Lagrange Multipliers with Set Valued Constraints by Using Contingent Epiderivatives

In this paper, we employ the generalized Guignard's constraint qualification to present the dual cone characterizations of the constraint set $S$ with set valued constraints in $\R^n.$ The obtained results provide sufficient conditions for which the ``strong conical hull intersection property`` (strong CHIP, in short) holds. Moreover, we establish necessary and sufficient conditions for characterizing ``perturbation property`` of the constrained best approximation to any point $x \in \R^{n}$ from a convex set $\tS:=K \cap S$ by the strong CHIP of $K$ and $S$ at a reference point, where $K$ is a non-empty closed convex set in $ \R^{n}.$ Finally, under the generalized Guignard's constraint qualification we derive the Lagrange multipliers characterizations of the constrained best approximation with set valued constraints. The clarification of our results is illustrated by the numerical experiments.