On generalized trade-off directions for basic optimality principles in convex and noncon- vex 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 tradeoff 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 optimality conditions in terms of corresponding tradeoff directions for both convex and nonconvex cases. We analyze similarities and differences between the cases.