Safeguarded augmented Lagrangian algorithms with scaled stopping criterion for the subproblems

At each iteration of the safeguarded augmented Lagrangian algorithm Algencan, a bound-constrained subproblem consisting of the minimization of the Powell–Hestenes–Rockafellar augmented Lagrangian function is considered, for which an approximate minimizer with tolerance tending to zero is sought. More precisely, a point that satisfies a subproblem first-order necessary optimality condition with tolerance tending to zero is required. In this work, based on the success of scaled stopping criteria in constrained optimization, we propose a scaled stopping criterion for the subproblems of Algencan. The scaling is done with the maximum absolute value of the first-order Lagrange multipliers approximation, whenever it is larger than one. The difference between the convergence theory of the scaled and non-scaled versions of Algencan is discussed and extensive numerical experiments are provided.