A Unified Separation Theorem for Closed Sets in a Banach Space and Optimality Conditions for Vector Optimization

Using the technique of variational analysis and in terms of normal cones, we establish unified separation results for finitely many closed (not necessarily convex) sets in Banach spaces, which not only cover the existing nonconvex separation results and a classical convex separation theorem but also recapture the approximate projection theorem. With help of the separation result for closed sets, we provide necessary and sufficient conditions for approximate Pareto solutions of constrained vector optimization problems. In particular, we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting.