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Volume 36, Issue 3
An Augmented Lagrangian Trust Region Method with a Bi-Object Strategy

Caixia Kou, Zhongwen Chen, Yuhong Dai & Haifei Han

DOI: 10.4208/jcm.1705-m2016-0820

J. Comp. Math., 36 (2018), pp. 331-350.

Published online: 2018-06

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  • Abstract

An augmented Lagrangian trust region method with a bi-object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.

  • Keywords

Nonlinear constrained optimization, Augmented Lagrangian function, Bi-object strategy, Global convergence.

  • AMS Subject Headings

90C55, 65K05, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

koucx@lsec.cc.ac.cn (Caixia Kou)

zwchen@suda.edu.cn (Zhongwen Chen)

dyh@lsec.cc.ac.cn (Yuhong Dai)

572852834@qq.com (Haifei Han)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-331, author = {Kou , CaixiaChen , ZhongwenDai , Yuhong and Han , Haifei}, title = {An Augmented Lagrangian Trust Region Method with a Bi-Object Strategy}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {3}, pages = {331--350}, abstract = {

An augmented Lagrangian trust region method with a bi-object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1705-m2016-0820}, url = {http://global-sci.org/intro/article_detail/jcm/12264.html} }
TY - JOUR T1 - An Augmented Lagrangian Trust Region Method with a Bi-Object Strategy AU - Kou , Caixia AU - Chen , Zhongwen AU - Dai , Yuhong AU - Han , Haifei JO - Journal of Computational Mathematics VL - 3 SP - 331 EP - 350 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1705-m2016-0820 UR - https://global-sci.org/intro/article_detail/jcm/12264.html KW - Nonlinear constrained optimization, Augmented Lagrangian function, Bi-object strategy, Global convergence. AB -

An augmented Lagrangian trust region method with a bi-object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.

Kou , CaixiaChen , ZhongwenDai , Yuhong and Han , Haifei. (2018). An Augmented Lagrangian Trust Region Method with a Bi-Object Strategy. Journal of Computational Mathematics. 36 (3). 331-350. doi:10.4208/jcm.1705-m2016-0820
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