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 a 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.