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A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization

Author

Listed:
  • Paul Armand

    (University of Limoges)

  • Riadh Omheni

    (SAS Institute Inc.)

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.

Suggested Citation

  • Paul Armand & Riadh Omheni, 2017. "A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 523-547, May.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-017-1071-x
    DOI: 10.1007/s10957-017-1071-x
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    References listed on IDEAS

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    1. Paul Armand & Joël Benoist & Riadh Omheni & Vincent Pateloup, 2014. "Study of a primal-dual algorithm for equality constrained minimization," Computational Optimization and Applications, Springer, vol. 59(3), pages 405-433, December.
    2. Philip Gill & Daniel Robinson, 2012. "A primal-dual augmented Lagrangian," Computational Optimization and Applications, Springer, vol. 51(1), pages 1-25, January.
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    Cited by:

    1. Paul Armand & Ngoc Nguyen Tran, 2021. "Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 96-116, April.
    2. Spyridon Pougkakiotis & Jacek Gondzio, 2019. "Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 905-945, June.
    3. Spyridon Pougkakiotis & Jacek Gondzio, 2021. "An interior point-proximal method of multipliers for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 78(2), pages 307-351, March.
    4. Jacek Gondzio & Spyridon Pougkakiotis & John W. Pearson, 2022. "General-purpose preconditioning for regularized interior point methods," Computational Optimization and Applications, Springer, vol. 83(3), pages 727-757, December.
    5. Nitish Das & P. Aruna Priya, 2019. "A Gradient-Based Interior-Point Method to Solve the Many-to-Many Assignment Problems," Complexity, Hindawi, vol. 2019, pages 1-13, July.
    6. Paul Armand & Ngoc Nguyen Tran, 2019. "An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 197-215, April.

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