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Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization

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Abstract

At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual convergence theories. However, in practice, one might not be able to solve the subproblem up to the required precision. This may be due to different reasons. One of them is that the presence of an excessively large penalty parameter could impair the performance of the box-constraint optimization solver. In this paper a practical strategy for decreasing the penalty parameter in situations like the one mentioned above is proposed. More generally, the different decisions that may be taken when, in practice, one is not able to solve the Augmented Lagrangian subproblem will be discussed. As a result, an improved Augmented Lagrangian method is presented, which takes into account numerical difficulties in a satisfactory way, preserving suitable convergence theory. Numerical experiments are presented involving all the CUTEr collection test problems.

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Authors and Affiliations

  1. Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil

    Ernesto G. Birgin

  2. Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, SP, Brazil

    J. M. Martínez

Authors
  1. Ernesto G. Birgin
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  2. J. M. Martínez
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Corresponding author

Correspondence to Ernesto G. Birgin.

Additional information

This work was supported by PRONEX-CNPq/FAPERJ (E-26/171.1510/2006-APQ1), FAPESP (2006/53768-0, 2006/03496-3 and 2009/10241-0) and CNPq (Grant 304484/2007-5).

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Birgin, E.G., Martínez, J.M. Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization. Comput Optim Appl 51, 941–965 (2012). https://doi.org/10.1007/s10589-011-9396-0

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