Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, and vanishing constraints. In addition to the tightness of the respective relaxations, we also address the issue of constraint qualification and the behavior of computational methods for some formulations. As a benchmark, we formulate a truck cruise control problem with logical implications resulting from gear-choice specific constraints. We provide this benchmark problem in AMPL format along with different realistic scenarios. Computational results for this benchmark are used to investigate feasibility gaps, integer feasibility gaps, quality of local solutions, and well-behavedness of numerical methods for the presented reformulations of the benchmark problem. Vanishing constraints give the most satisfactory results.
M. Juenger, G. Reinelt (eds). Festschrift fuer Martin Groetschel. Springer Verlag, 2013.