We study an optimization problem given by a discrete inclusion with end point constraints. An approach concerning second-order optimality conditions is proposed.

Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.

We prove the existence of a sequence $\left(x_k\right)$ satisfying $0\in f\left(x_k\right)+{\sum }_{i=1}^M{a}_i\nabla f(x_k+{\beta }_i(x_{k+1}-x_k))(x_{k+1}-x_k)+F\left(x_{k+1}\right)$, where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.