Some Second-order Characterizations For Benson Proper Efficient Solutions Of Set-valued Optimization

In this paper,the Benson proper efficient solutions of set-valued optimization problems are studied by means of the second-order tangent epiderivative.By using the separation theorem of convex sets,the necessary optimality conditions for the solutions are established.When the objective function and the constraint function are both cone convex,the sufficient optimality conditionsof the solutions are obtained.In the general mathematical model,because some secondary factors are ignored,the model is often approximate,and the solution obtained by numerical algorithm for the mathematical model is mostly approximate.On the other hand,when the feasible set is not compact,the set of exact solutions is usually empty,Under weaker conditions,the set approximate solutions can be nonempty.In Hausdorff locally convex topological linear space,the approximate Benson proper efficient solutions of unconstrained and constrained set-valued equilibrium problems are studied respectively.Not have any convexity hypothesis,the optimality conditions are established by using nonlinear functional.