On the paper “Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem”

In the paper Torrealba et al. (2021) an augmented Lagrangian algorithm was proposed for resource allocation problems with the intriguing characteristic that instead of solving the box-constrained augmented Lagrangian subproblem, they propose projecting the solution of the unconstrained subproblem onto such box. A global convergence result for the quadratic case was provided, however, this is somewhat counterintuitive, as in usual augmented Lagrangian theory, this strategy can fail in solving the augmented Lagrangian subproblems. In this note we investigate further this algorithm and we show that the proposed method may indeed fail when the Hessian of the quadratic is not a multiple of the identity. In the paper, it is not clear enough that two different projections are being used: one for obtaining their convergence results and other in their implementation. However, despite the lack of theoretical convergence, their strategy works remarkably well in some classes of problems; thus, we propose a hybrid method which uses their idea as a starting point heuristics, switching to a standard augmented Lagrangian method under certain conditions. Our contribution consists in presenting an efficient way of determining when the heuristics is failing to improve the KKT residual of the problem, suggesting that the heuristic procedure should be abandoned. Numerical results are provided showing that this strategy is successful in accelerating the standard method.