Second-order Characterizations On Benson Proper Efficient Element And Henig Proper Efficient Element Of Set-valued Optimization

A new kind of second-order tangent cone is introduced. With which a new kind of second-order tangent derivative, termed second-order radial composed tangent derivative, is introduced. Some properties of second-order radial composed tangent derivative and the relationship to second-order composed tangent derivative are discussed. With the help of second-order radial tangent derivative, sufficient and necessary optimality conditions are established for a Benson proper efficient element of set-valued optimization.A new second-order TP composed tangent cone is introduced by using the TP tangent cone, and a new second-order TP composed tangent derivative is given. At the same time, an example is given to show that the existence condition is weaker than that of the second-order composed tangent derivative. In the real normed linear space, optimality conditions for the Henig proper efficient element of set-valued optimization by using second-order TP composed epiderivative for its new definition.