A high-order augmented Lagrangian method with arbitrarily fast convergence

TL;DR

This paper introduces a high-order augmented Lagrangian method that achieves arbitrarily fast convergence rates for convex optimization problems with linear constraints, with applications across scientific fields.

Contribution

The paper develops a novel high-order augmented Lagrangian method with proven superlinear convergence, extending the proximal point framework under convexity assumptions.

Findings

Achieves superlinear convergence rates.

Applicable to data fitting, porous media flow, and scientific machine learning.

Demonstrates effectiveness through theoretical analysis and practical applications.

Abstract

We propose a high-order version of the augmented Lagrangian method for solving convex optimization problems with linear constraints, which achieves arbitrarily fast -- and even superlinear -- convergence rates. First, we analyze the convergence rates of the high-order proximal point method under certain uniform convexity assumptions on the energy functional. We then introduce the high-order augmented Lagrangian method and analyze its convergence by leveraging the convergence results of the high-order proximal point method. Finally, we present applications of the high-order augmented Lagrangian method to various problems arising in the sciences, including data fitting, flow in porous media, and scientific machine learning.