  This paper examines solution methods for mathematical programs with
complementarity constraints (MPCC) obtained from the time-discretization of
optimal control problems (OCPs) subject to nonsmooth dynamical systems. The
MPCC theory and stationarity concepts are reviewed and summarized. The focus is
on relaxation-based methods for MPCCs, which solve a (finite) sequence of more
regular nonlinear programs (NLP), where a regularization/homotopy parameter is
driven to zero. Such methods perform reasonably well on currently available
benchmarks. However, these results do not always generalize to MPCCs obtained
from nonsmooth OCPs. To provide a more complete picture, this paper introduces
a novel benchmark collection of such problems, which we call nosbench. The
problem set includes 603 different MPCCs and we split it into a few
representative subsets to accelerate the testing. We compare different
relaxation-based methods, NLP solvers, homotopy parameter update and relaxation
parameter steering strategies. Moreover, we check whether the obtained
stationary points allow first-order descent directions, which may be the case
for some of the weaker MPCC stationarity concepts. In the best case, the
Scholtes' relaxation [Scholtes, 2002] with IPOPT [W\"achter and Biegler, 2006]
as NLP solver manages to solve 73.8 % of the problems. This highlights the need
for further improvements in algorithms and software for MPCCs.
