  Least-squares programming is a popular tool in robotics due to its simplicity
and availability of open-source solvers. However, certain problems like sparse
programming in the $\ell_0$- or $\ell_1$-norm for time-optimal control are not
equivalently solvable. In this work, we propose a non-linear hierarchical
least-squares programming (NL-HLSP) for time-optimal control of non-linear
discrete dynamic systems. We use a continuous approximation of the heaviside
step function with an additional term that avoids vanishing gradients. We use a
simple discretization method by keeping states and controls piece-wise constant
between discretization steps. This way, we obtain a comparatively easily
implementable NL-HLSP in contrast to direct transcription approaches of optimal
control. We show that the NL-HLSP indeed recovers the discrete time-optimal
control in the limit for resting goal points. We confirm the results in
simulation for linear and non-linear control scenarios.
