Generalized Cone-Arcwise Connected Set-valued Mapping And Applications

The concept of ?- order nearly con?-arcwise connected set-valued mapping is introduced. An example is given to illustrate that the ?- order nearly con?-arcwise connected set-valued mapping is a proper generalization of the cone arcwise connected set-valued mapping. With the help of Y- contingent cone, generalized Y- contingent epiderivative is introduced, and the relationship between generalized Y- contingent epiderivative and generalized contingent epiderivative is discussed. When objective function is ?- order nearly con?-arcwise connected, for weakly efficient elements optimality sufficient and necessary conditions are also established.The concept of ?- super subgradient for a set-valued map is introduced. Under certain hypothesis, by applying separation theorem for convex sets, the existence theorem for an ?- super subgradient is obtained. An example is given to illustrate the main result.The concept of lower radial contingent derivative for set-valued map is introduced with help of lower radial contingent cone. By using this concept, some important properties are presented. An optimality necessary condition for a point pair to be a global proper efficient element of set-valued optimization problem is established where objective function and constraint function are separated.