A numerical study of transformed mixed-integer optimal control problems

Time transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal control problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. The approach is well understood and used widely due to its many favorable properties. Recently, several extensions and algorithmic improvements have been proposed. We use an alternative formulation, the partial outer convexification (POC), to study convergence properties of (STO). We introduce the open-source software package ampl_mintoc (Sager et al., czeile/ampl_mintoc: Math programming c release, 2024, https://doi.org/10.5281/zenodo.12520490). It is based on AMPL, designed for the formulation of mixed-integer optimal control problems, and allows to use almost identical implementations for (STO) and (POC). We discuss and explain our main numerical result: (STO) is likely to result in more local minima for each discretization grid than (POC), but the number of local minima is asymptotically identical for both approaches.