The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously.