Optimality Conditions for Disjunctive Optimization in Reflexive Banach Spaces

We study optimization problems with the constraints having disjunctive structures in reflexive Banach spaces. By the representations of contingent cones and Fr,chet normal cones to finite unions of sets in general Banach spaces and using the special structures of convex generalized polyhedral sets, we calculate the Mordukhovich normal cones to finite unions of closed and convex sets that particularly include convex generalized polyhedral sets in reflexive Banach spaces. Furthermore, based on these calculations and the Guignard-type constraint qualifications, we derive new optimality conditions for disjunctive optimization problems. We also present specializations of these results to optimization problems with variational inequality constraints.