  We present a numerical method for the minimization of objectives that are
augmented with large quadratic penalties of overdetermined inconsistent
equality constraints. Such objectives arise from quadratic integral penalty
methods for the direct transcription of equality constrained 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 of the discretization 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 explore a modification
of ALM that fits our purpose. 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.
