A semismooth Newton augmented Lagrangian method for solving nonlinear programming

The augmented Lagrangian method (ALM) has been widely used to solve nonlinear programming problems. Although there has been extensive research in the algorithmic aspect, the box constraints are typically preserved during the minimization of the augmented Lagrangian function. In this paper, we propose an equivalent reformulation of the problem by splitting the variables in the constraint. We then establish the augmented Lagrangian function based on projection operators to propose an ALM framework, where the subproblem is an unconstrained optimization problem. For the ALM subproblem, we utilize efficient techniques to approximate the Hessian matrix of the augmented Lagrangian function and design an inexact semismooth Newton method as the solver. We also provide convergence and complexity properties of the algorithm. Finally, we conduct numerical experiments and comparisons with the interior point and sequential quadratic programming methods on nonlinear programming instances from the CUTEr (constrained and unconstrained testing environment) dataset. The results demonstrate the effectiveness of the proposed ALM.