  Motivated by robotic trajectory optimization problems we consider the
Augmented Lagrangian approach to constrained optimization. We first propose an
alternative augmentation of the Lagrangian to handle the inequality case (not
based on slack variables) and a corresponding "central" update of the dual
parameters. We proove certain properties of this update: roughly, in the case
of LPs and when the "constraint activity" does not change between iterations,
the KKT conditions hold after just one iteration. This gives essential insight
on when the method is efficient in practise. We then present our main
contribution, which are consistent any-time (non-central) updates of the dual
parameters (i.e., updating the dual parameters when we are not currently at an
extremum of the Lagrangian). Similar to the primal-dual Newton method, this
leads to an algorithm that parallely updates the primal and dual solutions, not
distinguishing between an outer loop to adapt the dual parameters and an inner
loop to minimize the Lagrangian. We again proof certain properties of this
anytime update: roughly, in the case of LPs and when constraint activities
would not change, the dual solution converges after one iteration. Again, this
gives essential insight in the caveats of the method: if constraint activities
change the method may destablize. We propose simple smoothing, step-size
adaptation and regularization mechanisms to counteract this effect and
guarantee monotone convergence. Finally, we evaluate the proposed method on
random LPs as well as on standard robot trajectory optimization problems,
confirming our motivation and intuition that our approach performs well if the
problem structure implies moderate stability of constraint activity.
