Generalizing trade-off deirections in multiobjective optimization

We consider a general multiobjective optimization problem with five basic optimality principles: efficiency, weak and proper Pareto optimality, strong efficiency and lexicographic optimality. We generalize the concept of trade-off directions defining them as some optimal surface of appropriate cones. In convex optimization, the contingent cone can be used for all optimality principles except lexicographic optimality, where the cone of feasible directions is useful. In nonconvex case the contingent cone and the cone of locally feasible directions with lexicographic optimality are helpful. We derive necessary and sufficient geometrical optima lity conditions in terms of corresponding trade-off directions for both convex and nonconvex cases.