Modified Augmented Lagrangian Method for the minimization of functions with quadratic penalty terms

TL;DR

This paper introduces a modified Augmented Lagrangian Method tailored for efficiently minimizing functions with large quadratic penalty terms, extending its applicability beyond traditional equality-constrained problems.

Contribution

The paper proposes a novel modification of the ALM that effectively handles unconstrained problems with large quadratic penalties, broadening its practical utility.

Findings

The modified ALM converges efficiently for penalty-heavy problems.

The method maintains the exact constraint satisfaction property.

It outperforms traditional quadratic penalty approaches in convergence speed.

Abstract

The Augmented Lagrangian Method (ALM) is an iterative method for the solution of equality-constrained non-linear programming problems. In contrast to the quadratic penalty method, the ALM can satisfy equality constraints in an exact way. Further, ALM is claimed to converge in less iterations, indicating that it is superior in approach to a quadratic penalty method. It is referred to as an advantage that the ALM solves equality constraints in an exact way, meaning that the penalty parameter does not need to go to infinity in order to yield accurate feasibility for the constraints. However, we sometimes actually want to minimize an unconstrained problem that has large quadratic penalty terms. In these situations it is unclear how the ALM could be utilized in the correct way. This paper presents the answer: We derive a modified version of the ALM that is also suitable for solving functions with large quadratic penalty terms.