Note on the Modifed Augmented Lagrangian Method for Minimization of Functions with Large Quadratic Penalties

TL;DR

This paper presents a reformulation of the Modified Augmented Lagrangian Method (MALM) for more stable and efficient minimization of functions with large quadratic penalties, replacing the inner Quasi-Newton iteration with a well-scaled unconstrained minimization approach.

Contribution

It introduces a new formulation of MALM using a Newton iteration-based unconstrained minimization, improving numerical stability and efficiency over the previous Quasi-Newton approach.

Findings

Reformulation enhances numerical stability.

Unconstrained minimization simplifies implementation.

Relation between MALM and ALM is clarified.

Abstract

In a recent work (arXiv-DOI: 1804.08072v1) we introduced the Modified Augmented Lagrangian Method (MALM) for the efficient minimization of objective functions with large quadratic penalty terms. From MALM there results an optimality equation system that is related to that of the original objective function. But, it is numerically better behaved, as the large penalty factor is replaced by a milder factor. In our original work, we formulated MALM with an inner iteration that applies a Quasi-Newton iteration to compute the root of a multi-variate function. In this note we show that this formulation of the scheme with a Newton iteration can be replaced conveniently by formulating a well-scaled unconstrained minimization problem. In this note, we briefly review the Augmented Lagrangian Method (ALM) for minimizing equality-constrained problems. Then we motivate and derive the new proposed formulation of MALM for minimizing unconstrained problems with large quadratic penalties. Eventually, we discuss relations between MALM and ALM.