Enhancing sharp augmented Lagrangian methods with smoothing techniques for nonlinear programming

TL;DR

This paper introduces a smoothing-enhanced sharp augmented Lagrangian method for nonlinear programming, improving computational efficiency and robustness by approximating nonconvex functions, and demonstrating favorable numerical performance.

Contribution

It develops a novel smoothing technique for sharp augmented Lagrangian methods, enabling primal stationarity and practical efficiency in solving complex nonlinear programming problems.

Findings

Smoothing approach improves robustness over traditional methods

Algorithms perform well in numerical experiments

Method balances theoretical rigor and computational feasibility

Abstract

This paper proposes a novel approach to solving nonlinear programming problems using a sharp augmented Lagrangian method with a smoothing technique. Traditional sharp augmented Lagrangian methods are known for their effectiveness but are often hindered by the need for global minimization of nonconvex, nondifferentiable functions at each iteration. To address this challenge, we introduce a smoothing function that approximates the sharp augmented Lagrangian, enabling the use of primal minimization strategies similar to those in Powell--Hestenes--Rockafellar (PHR) methods. Our approach retains the theoretical rigor of classical duality schemes while allowing for the use of stationary points in the primal optimization process. We present two algorithms based on this method--one utilizing standard descent and the other employing coordinate descent. Numerical experiments demonstrate that our smoothing--based method compares favorably with the PHR augmented Lagrangian approach, offering both robustness and practical efficiency. The proposed method is particularly advantageous in scenarios where exact minimization is computationally infeasible, providing a balance between theoretical precision and computational tractability.