  In this work, we present several tools for efficient sequential hierarchical
least-squares programming (S-HLSP) for lexicographical optimization tailored to
robot control and planning. As its main step, S-HLSP relies on approximations
of the original non-linear hierarchical least-squares programming (NL-HLSP) to
a hierarchical least-squares programming (HLSP) by the hierarchical Newton's
method or the hierarchical Gauss-Newton algorithm. We present a threshold
adaptation strategy for appropriate switches between the two. This ensures
optimality of infeasible constraints, promotes numerical stability when solving
the HLSP's and enhances optimality of lower priority levels by avoiding
regularized local minima. We introduce the solver $\mathcal{N}$ADM$_2$, an
alternating direction method of multipliers for HLSP based on nullspace
projections of active constraints. The required basis of nullspace of the
active constraints is provided by a computationally efficient turnback
algorithm for system dynamics discretized by the Euler method. It is based on
an upper bound on the bandwidth of linearly independent column subsets within
the linearized constraint matrices. Importantly, an expensive initial
rank-revealing matrix factorization is unnecessary. We show how the high
sparsity of the basis in the fully-actuated case can be preserved in the
under-actuated case. $\mathcal{N}$ADM$_2$ consistently shows faster
computations times than competing off-the-shelf solvers on NL-HLSP composed of
test-functions and whole-body trajectory optimization for fully-actuated and
under-actuated robotic systems. We demonstrate how the inherently lower
accuracy solutions of the alternating direction method of multipliers can be
used to warm-start the non-linear solver for efficient computation of high
accuracy solutions to non-linear hierarchical least-squares programs.
