Optimality conditions for convex problems on intersections of non necessarily convex sets

We present necessary and sufficient optimality conditions for the minimization of pseudoconvex functions over convex intersections of non necessarily convex sets. To this aim, we use the notion of local normal cone to a closed set at a point, due to Linh and Penot (SIAM J Optim 17:500–510, 2006). The technique we use to obtain the optimality conditions is based on the so called canonical representation of a closed set by means of its associated oriented distance function.