Nonconvex Multiobjective Programming: Geometry of Optimality via Cones

Various type of optimal solutions of multiobjective optimization problems can be characterized by means of different cones. We consider here five different optimality principles which are very common in multiobjective optimization: efficiency, weak and proper Pareto optimality, strong and lexicographic optimality. The five optimality concepts can be characterized with the help of different geometrical concepts. The usage of contingent cone, normal cone and cone of feasible directions is a natural choice in the case of convex optimization. In nonconvex case two additional types of cones are helpful tangent cone and cone of local feasible directions. Provided the partial objectives are not necessarily convex, we derive necessary and sufficient geometrical optimality conditions for strongly efficient and lexicographically optimal solutions by using the above-mentioned cones. Combining new results with previously known ones about efficiency, weak and proper Pareto optimality, we derive two general schemes reflecting structural properties and interconnections of the five optimality principles.

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