Recent advances in trust region algorithms

Abstract

Trust region methods are a class of numerical methods for optimization. Unlike line search type methods where a line search is carried out in each iteration, trust region methods compute a trial step by solving a trust region subproblem where a model function is minimized within a trust region. Due to the trust region constraint, nonconvex models can be used in trust region subproblems, and trust region algorithms can be applied to nonconvex and ill-conditioned problems. Normally it is easier to establish the global convergence of a trust region algorithm than that of its line search counterpart. In the paper, we review recent results on trust region methods for unconstrained optimization, constrained optimization, nonlinear equations and nonlinear least squares, nonsmooth optimization and optimization without derivatives. Results on trust region subproblems and regularization methods are also discussed.

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Acknowledgments

I am very grateful to my former students Jinyan Fan, Yong Xia, Zaikun Zhang and Xiao Wang for their helps during my preparation of this paper. I would like to thank my long-term colleagues and dear friends Philippe Toint and Andew Conn, and two anonymous referees for their valuable comments which help to improve the paper. This paper is supported in partial by Grants 11331012, 11321061 and 11461161005 of the National Natural Science Foundation of China.

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Correspondence to Ya-xiang Yuan.

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Yuan, Y. Recent advances in trust region algorithms. Math. Program. 151, 249–281 (2015). https://doi.org/10.1007/s10107-015-0893-2

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Keywords

  • Trust region algorithms
  • Nonlinear optimization
  • Subproblem
  • Complexity
  • Convergence

Mathematics Subject Classification

  • 65K05
  • 90C30