A Novel Augmented Lagrangian Approach for Inequalities and Convergent   Any-Time Non-Central Updates
  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. 
Marc Toussaint
math.OC