In this paper, we utilize the improvement set and the recession cone to introduce the concept of \(E_{\infty }\) -Benson properly efficient solution of the set-valued equilibrium problems and set-valued optimization problems. We establish scalarization optimality conditions, both linear and nonlinear, for \(E_{\infty }\) -Benson proper efficiency of the set-valued equilibrium problems and also of the set-valued optimization problems. Based on the linear scalarization, we derive Lagrange multiplier optimality conditions for \(E_{\infty }\) -Benson proper efficiency of the set-valued equilibrium problems and also of the set-valued optimization problems. Several results obtained in this paper are illustrated by examples. The results obtained in this paper improve and generalize some known results in the literature.
