Augmented Lagrangian Method for Linear Programming Using Smooth Approximation

The augmented Lagrangian method can be used for finding the least \(2-\) norm solution of a linear programming problem. This approach’s primary advantage is that it leads to the minimization of an unconstrained problem with a piecewise quadratic, convex, and differentiable objective function. However, this function lacks an ordinary Hessian, which precludes the use of a fast Newton method. In this paper, we apply the smoothing techniques and solve an unconstrained smooth reformulation of this problem using a fast Newton method. Computational results and comparisons are illustrated through multiple numerical examples to show the effectiveness of the proposed algorithm.