Solving Problems with Inconsistent Constraints with a Modified Augmented Lagrangian Method

TL;DR

This paper introduces a modified Augmented Lagrangian Method to effectively solve constrained optimization problems with inconsistent constraints, outperforming traditional methods in convergence speed and accuracy.

Contribution

The paper proposes a new modification to the ALM that ensures convergence even with inconsistent constraints and provides theoretical convergence guarantees.

Findings

Modified ALM converges faster than QPM in numerical tests.

Modified ALM minimizes quadratic penalty-augmented functions effectively.

Unmodified ALM may converge to a different problem's minimizer.

Abstract

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.