A modified augmented lagrangian method for problems with inconsistent constraints

Abstract

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.