The augmented Lagrangian-based splitting methods have found more and more applications in scientific and engineering computation, such as compressive sensing, covariance selection, image processing, and transportation research. One of the basic difficulties in such algorithms is the selection of the parameter in the augmented Lagrangian function; a larger one may make the primal progress too small, while a smaller one may slow down the dual progress. To overcome this difficulty, in this paper, we propose to solve the splitting subproblems in a trust region manner, and the radius can be adjusted smartly. Under the same mild conditions as those for classical augmented Lagrangian-based splitting methods, we prove the global convergence of the proposed algorithm. Moreover, the \(\mathcal {O}(1/\epsilon )\) convergence rate is also analyzed in an ergodic sense. We present some preliminary numerical experiments on medical image recovery and logistic regression, which show that the trust region-based splitting method is efficient and promising.

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A Trust-Region-Based Splitting Method for Optimization Problems with Linear Constraints

  • Leyu Hu,
  • Yannan Chen,
  • Xingju Cai,
  • Deren Han

摘要

The augmented Lagrangian-based splitting methods have found more and more applications in scientific and engineering computation, such as compressive sensing, covariance selection, image processing, and transportation research. One of the basic difficulties in such algorithms is the selection of the parameter in the augmented Lagrangian function; a larger one may make the primal progress too small, while a smaller one may slow down the dual progress. To overcome this difficulty, in this paper, we propose to solve the splitting subproblems in a trust region manner, and the radius can be adjusted smartly. Under the same mild conditions as those for classical augmented Lagrangian-based splitting methods, we prove the global convergence of the proposed algorithm. Moreover, the \(\mathcal {O}(1/\epsilon )\) convergence rate is also analyzed in an ergodic sense. We present some preliminary numerical experiments on medical image recovery and logistic regression, which show that the trust region-based splitting method is efficient and promising.