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

TL;DR

This paper introduces a modified Augmented Lagrangian Method (ALM) for efficiently solving integral penalty transcriptions of optimal control problems, especially when constraints are inconsistent, outperforming traditional quadratic penalty methods.

Contribution

The paper proposes a modification to the ALM that improves convergence and speed for solving integral penalty formulations of optimal control problems with inconsistent constraints.

Findings

Modified ALM converges faster than QPM in numerical experiments.

Unmodified ALM converges to a different minimizer, not the intended one.

Modified ALM effectively minimizes quadratic penalty-augmented functions.

Abstract

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.