Second-Order Modified Contingent Epiderivatives in Set-Valued Optimization

In this article, we introduce a second-order modified contingent cone and a second-order modified contingent epiderivative. We discuss some properties of the second-order cone and the epiderivative, respectively. Moreover, a Fritz John type necessary optimality condition is obtained for the set-valued optimization problems with constraints by using the second-order modified contingent epiderivative and an example is proposed to explain the Fritz John type necessary optimality condition. In particular, we obtain a unified second-order sufficient and necessary optimality condition for the set-valued optimization problems with constraints under twice differentiable L-quasi-convex assumption.