In this paper, we introduce new nonlinear scalarization methods to characterize minimal elements of set-valued problems equipped with variable domination structures. For certain set relations with respect to variable domination structures, we give a characterization of these relations using corresponding nonlinear scalarizing functionals and the properties of these functionals. We describe different kinds of minimal elements to a family of sets with respect to variable domination structures. Furthermore, we investigate a descent method to find approximations of minimal solutions of set-valued problems with respect to variable domination structures using nonlinear scalarization methods. Finally, we apply the results to a set-valued problem in Medical Image Registration.