Higher-Order Radial Hadamard Directional Derivatives and Applications to Set-Valued Equilibrium Problems

In this paper, we first introduce the higher-order lower and upper radial Hadamard directional derivatives for set-valued maps, which can capture the global information of concerning maps, and discuss the relationships with other existed higher-order derivatives. Second, based on the introduced derivatives, we establish optimality conditions for Henig efficient solutions of a set-valued equilibrium problem with constraints. Particularly, the optimality conditions hold in the case where the derivatives of objective and constraint functions are separated. Finally, we give some duality theorems for a mixed type of primal-dual set-valued equilibrium problem. The main results of this paper are illustrated by several concrete examples.