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


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.


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


90C20 90C06 65F10 65F15 65F35 


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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).


  1. 1.
    Conn A R, Gould N I M, Toint P L. Trust-Region Methods. Philadelphia: SIAM, 2000CrossRefzbMATHGoogle Scholar
  2. 2.
    Demmel J. Applied Numerical Linear Algebra. Philadelphia: SIAM, 1997CrossRefzbMATHGoogle Scholar
  3. 3.
    Gay D M. Computing optimal locally constrained steps. SIAM J Sci Statist Comput, 1981, 2: 186–197MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Golub G H, Van Loan C F. Matrix Computations. 3rd ed. Baltimore: Johns Hopkins University Press, 1996zbMATHGoogle Scholar
  5. 5.
    Golub G H, von Matt U. Quadratically constrained least squares and quadratic problems. Numer Math, 1991, 59: 561–580MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gould N I M, Lucidi S, Roma M, Toint P L. Solving the trust-region subproblem using the Lanczos method. SIAM J Optim, 1999, 9: 504–525MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hager W W. Minimizing a quadratic over a sphere. SIAM J Optim, 2001, 12: 188–208MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hager W W, Krylyuk Y. Graph partitioning and continuous quadratic programming. SIAM J Discrete Math, 1999, 12(4): 500–523MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li C K, Li R C. A note on eigenvalues of perturbed Hermitian matrices. Linear Algebra Appl, 2005, 395: 183–190MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li R C. On Meinardus’ examples for the conjugate gradient method. Math Comp, 2008, 77(261): 335–352MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li R C. Sharpness in rates of convergence for symmetric Lanczos method. Math Comp, 2010, 79(269): 419–435MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li R C, Zhang L H. Convergence of block Lanczos method for eigenvalue clusters. Numer Math, 2015, 131: 83–113MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lukšan L, Matonoha C, Vlček J. On Lagrange multipliers of trust-region subproblems. BIT, 2008, 48: 763–768MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moré J, Sorensen D C. Computing a trust region step. SIAM J Sci Statist Comput, 1983, 4(3): 553–572MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nocedal J, Wright S. Numerical Optimization. 2nd ed. Berlin: Springer, 2006zbMATHGoogle Scholar
  16. 16.
    Parlett B N. The Symmetric Eigenvalue Problem. Philadelphia: SIAM, 1998CrossRefzbMATHGoogle Scholar
  17. 17.
    Rendl R, Wolkowicz H. A semidefinite framework for trust region subproblems with applications to large scale minimization. Math Program, 1997, 77(2): 273–299MathSciNetzbMATHGoogle Scholar
  18. 18.
    Rojas M, Santos S A, Sorensen D C. Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization. ACM Trans Math Software, 2008, 34(2): 1–28MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rojas M, Santos S A, Sorensen D C. A new matrix-free algorithm for the large-scale trust-region subproblem. SIAM J Optim, 2000, 11: 611–646MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rojas M, Sorensen D C. A trust-region approach to the regularization of large-scale discrete forms of ill-posed problems. SIAM J Sci Comput, 2002, 23: 1843–1861MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saad Y. Iterative Methods for Sparse Linear Systems. 2nd ed. Philadelphia: SIAM, 2003CrossRefzbMATHGoogle Scholar
  22. 22.
    Sorensen D C. Minimization of a large-scale quadratic function subject to a spherical constraint. SIAM J Optim, 1997, 7: 141–161MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Steihaug T. The conjugate gradient method and trust regions in large scale optimization. SIAM J Numer Anal, 1983, 20: 626–637MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tarantola A. Inverse Problem Theory. Amsterdam: Elsevier, 1987zbMATHGoogle Scholar
  25. 25.
    Tikhonov A N. Regularization of incorrectly posed problems. Soviet Math, 1963, 4: 1624–1627zbMATHGoogle Scholar
  26. 26.
    Toint P L. Towards an efficient sparsity exploiting Newton method for minimization. In: Duff I, ed. Sparse Matrices and Their Uses. London: Academic Press, 1981, 57–88Google Scholar
  27. 27.
    Zhang L H, Shen C G, Li R C. On the generalized Lanczos trust-region method. SIAM J Optim, 2017, 27(3): 2110–2142MathSciNetCrossRefzbMATHGoogle Scholar

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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