<p>In this paper, we aim to apply an oriented distance function to investigate minimal and weak minimal solutions of set optimization with a variable structure. We investigate some properties of the oriented distance function of type sup-inf with a variable structure. Moreover, we use this function to characterize the set relation with respect to a variable structure and obtain properties of the Dini directional derivatives for set-valued mappings. By using Dini directional derivatives, necessary and sufficient optimality conditions for the solutions are established, respectively. We show the separation between two suitable sets in image space to obtain the optimality conditions for constrained set optimization and introduce a generalized Lagrangian set-valued mapping to obtain the relationship between the saddle point and the existence of a nonlinear separation. Finally, as an application, we examine medical image registration.</p>