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Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 459–481 | Cite as

Error bounds of Lanczos approach for trust-region subproblem

  • Leihong Zhang
  • Weihong Yang
  • Chungen Shen
  • Jiang Feng
Research Article

Abstract

Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

Keywords

Trust-region method trust-region subproblem (TRS) Lanczos method Steihaug–Toint conjugate-gradient iteration error bound 

MSC

90C20 90C06 65F10 65F15 65F35 

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Notes

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and comments. This work of the first author was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671246, 91730303, 11371102) and the work of the second author was supported in part by the National Natural Science Foundation of China (Grant Nos. 91730304, 11371102, 91330201).

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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Leihong Zhang
    • 1
    • 2
  • Weihong Yang
    • 3
  • Chungen Shen
    • 4
  • Jiang Feng
    • 1
  1. 1.School of MathematicsShanghai University of Finance and EconomicsShanghaiChina
  2. 2.Shanghai Key Laboratory of Financial Information TechnologyShanghai University of Finance and EconomicsShanghaiChina
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiChina
  4. 4.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiChina

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