In this paper, we consider a set optimization problem with a partial order relation, which is defined by Minkowski difference. By using the image space analysis, we establish the relationships among the set optimization problem, a vector optimization problem and a set-valued optimization with vector criterion related to the image of the set optimization problem. In addition, two nonlinear regular weak separation functions are proposed for the set optimization problem. Based on the two nonlinear regular weak separation functions, saddle point sufficient optimality conditions, gap functions and error bounds for the set optimization problem, are obtained. Finally, we explore some applications of the obtained results to investigate robust multi-objective optimization problems and verify the validity of the results in shortest path problems with data uncertainty and multi-criteria traffic network equilibrium problems with interval-valued cost functions.