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A Direct Method for Solving Integral Penalty Transcriptions of Optimal Control Problems

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.