Optimality of Borwein robust properly efficient solutions for an uncertain multi-objective optimization problem via vector separation functions

This paper is devoted to establish optimality conditions of Borwein robust properly efficient solutions for an uncertain multi-objective optimization problem. Firstly, Borwein robust proper efficiency is proposed by an improvement set, and we emphasize the differences between it and the existed solutions (such as classical Borwein properly minimal solutions and optimal solutions). Secondly, we put forward a class of vector-valued functions to characterize the nonlinear separation between two sets in image spaces, and obtain its property. Finally, two saddle point type optimality theorems for the uncertain multi-objective optimization problem are derived based on the introduced separation functions.