We present a numerical method for the minimization of objectives that are
augmented with linear inequality constraints and large quadratic penalties of
over-determined inconsistent equality constraints. Such objectives arise from
quadratic integral penalty methods for the direct transcription of optimal
control problems.
The Augmented Lagrangian Method (ALM) has a number of advantages over the
Quadratic Penalty Method (QPM) for solving this class of problems. However, if
the equality constraints are inconsistent, then ALM might not converge to a
point that minimizes the %unconstrained bias of the objective and penalty term.
Therefore, in this paper we show a modification of ALM that fits our purpose.
We prove convergence of the modified method and prove under local uniqueness
assumptions that the local rate of convergence of the modified method in
general exceeds the one of the unmodified method.
Numerical experiments demonstrate that the modified ALM can minimize certain
quadratic penalty-augmented functions faster than QPM, whereas the unmodified
ALM converges to a minimizer of a significantly different problem.