  We present a numerical method for the minimization of constrained
optimization problems where the objective is augmented with large quadratic
penalties of 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). However, if the equality
constraints are inconsistent, then ALM might not converge to a point that
minimizes the bias of the objective and penalty term. Therefore, we present a
modification of ALM that fits our purpose. We prove convergence of the modified
method and bound its local convergence rate by that 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.
