Numerical comparison of collocation vs quadrature penalty methods

Direct transcription with collocation-type methods (CTM) is a popular approach for solving dynamic optimization problems. It is known that these types of methods can fail to converge for problems that feature singular-arc solutions, high-index differential-algebraic equations and over-determined constraints. Recently, we proposed the use of quadrature penalty methods (QPM) as an alternative numerical approach to collocation-type methods. In contrast to the concept of collocation, which requires constraint-residuals to equal zero at individual points (e.g. at collocation points), the main idea of QPM is to simply oversample this number of points and use their respective quadrature weights in a quadratic penalty term, coining the name of quadrature penalty. In this paper, we provide numerical case studies and a broad numerical comparison on a wide range of problems, highlighting the benefits of QPM over CTM not only in difficult problems, but also in solving problems competitively to CTM. These results show that QPM can be considered an attractive first go-to method when solving general dynamic optimization problems