Upper semicontinuity of closed-convex-valued multifunctions

In this paper we study the (Berge) upper semicontinuity of a generic multifunction assigning to each parameter, in a metric space, a closed convex subset of the n-dimensional Euclidean space. A relevant particular case arises when we consider the feasible set mapping associated with a parametric family of convex semi-infinite programming problems. Related to such a generic multifunction, we introduce the concept of epsilon-reinforced mapping, which will allow us to establish a sufficient condition for the aimed property. This condition turns out to be also necessary in the case that the boundary of the image set at the nominal value of the parameter contains no half-lines. On the other hand, it is well-known that every closed convex set in the Euclidean space can be viewed as the solution set of a linear semi-infinite inequality system and, so, a parametric family of linear semi-infinite inequality systems can always be associated with the original multifunction. In this case, a different necessary condition is provided in terms of the coefficients of these linear systems. This condition tries to measure the relative variation of the right hand side with respect to the left hand side of the constraints of the systems in a neighbourhood of the nominal parameter.