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A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization

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Abstract

We present a primal–dual algorithm for solving a constrained optimization problem. This method is based on a Newtonian method applied to a sequence of perturbed KKT systems. These systems follow from a reformulation of the initial problem under the form of a sequence of penalized problems, by introducing an augmented Lagrangian for handling the equality constraints and a log-barrier penalty for the inequalities. We detail the updating rules for monitoring the different parameters (Lagrange multiplier estimate, quadratic penalty and log-barrier parameter), in order to get strong global convergence properties. We show that one advantage of this approach is that it introduces a natural regularization of the linear system to solve at each iteration, for the solution of a problem with a rank deficient Jacobian of constraints. The numerical experiments show the good practical performances of the proposed method especially for degenerate problems.

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Acknowledgements

The authors would like to thank Dr. Joshua Griffin for his comments and suggestions which helped us to improve the presentation of the paper.

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Authors and Affiliations

  1. XLIM Laboratory - UMR CNRS no 7252, University of Limoges, 123, avenue Albert Thomas, 87060, Limoges, France

    Paul Armand

  2. Advanced Analytics Devision, Operations Research and Management Science R&D, SAS Institute Inc., Domaine de Grégy, 77257, Brie-Comte-Robert, France

    Riadh Omheni

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  1. Paul Armand
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  2. Riadh Omheni
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Correspondence to Paul Armand.

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Armand, P., Omheni, R. A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization. J Optim Theory Appl 173, 523–547 (2017). https://doi.org/10.1007/s10957-017-1071-x

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  • DOI: https://doi.org/10.1007/s10957-017-1071-x

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