Tangent and Normal Cones in Nonconvex Multiobjective Optimization

Abstract

Abstract. Trade-off information is important in multiobjective optimization. It describes the relationships of changes in objective function values. For example, in interactive methods we need information about the local behavior of solutions when looking for improved search directions.Henig and Buchanan have generalized in Mathematical Programming 7S(3), 1997 the concept of trade-offs in convex multiobjective optimization problems. With the help of tangent cones they define a cone of trade-off directions.In this paper, we examine the possibility of extending the results of Henig and Buchanan for nonconvex multiobjective optimization problems. We carry out the generalization in the sense of Clarke's nonconvex analysis.

Clarke's Local Generalized Nash Equilibria with Nonconvex Coupling Constraints

“…Remark 1: Whenever the feasible set X in (1) is convex, the CL-GNEP is equivalent to the GNEP, since the set of CL-GNEs is equal to that of GNEs. Indeed, Clarke's tangent cone of a convex set includes the convex set itself [26]. ■…”

“…In the related literature, several different definitions of cones have been proposed, hence, for the sake of clarity, let us here recall some concepts related to tangent cones following the convention used in [26] and [49]. Definition 6 (Clarke's tangent vector): Let us consider a nonempty subset X ⊆ R n , and a point x ∈ cl(X ).…”