Abstract:

High-level decision-making for dynamical systems often involves performance and safety specifications that are activated or deactivated depending on conditions related to the system state and commands. Such decisionmaking problems can be naturally formulated as optimization problems where these conditional activations are regulated by discrete variables. However, solving these problems can be challenging numerically, even on powerful computing platforms, especially when the dynamics are nonlinear. In this work, we consider decision-making for nonlinear systems where certain constraints, as well as possible terms in the cost function, are activated or deactivated depending on the system state and commands. We show that these problems can be formulated either as mixed-integer nonlinear programs (MINLPs) or as mathematical programs with vanishing constraints (MPVCs), where the former formulation involves discrete decision variables, whereas the latter relies on continuous variables subject to structured nonconvex constraints. We discuss the different solution methods available for both formulations and demonstrate them on optimal trajectory planning problems in various aerospace applications. Finally, we compare the strengths and weaknesses of the MINLP and MPVC approaches through a focused case study on powered descent guidance with divert-feasible regions. In our simulations for problems up to medium size, MPVC formulations provide accurate solutions faster than MINLP formulations. However, for larger problems, the MPVC formulation introduces numerous nonconvexities that hinder solver convergence, even when they are relatively simple, making MINLPs the preferred choice in such cases.