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 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.

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