×

zbMATH — the first resource for mathematics

On the generalized Lanczos trust-region method. (English) Zbl 1380.90210

MSC:
90C20 Quadratic programming
90C06 Large-scale problems in mathematical programming
65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide, 3rd ed., Software Environ. Tools, SIAM, Philadelphia, 1999.
[2] S. N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné., Mém. acad. royale Belg., 4 (1912), pp. 1–104.
[3] E. W. Cheney, Introduction to Approximation Theory, 2nd ed., Chelsea Publishing, New York, 1982. · Zbl 0535.41001
[4] A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods, MOS-SIAM Ser. Optim., SIAM, Philadelphia, 2000.
[5] J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997. · Zbl 0879.65017
[6] E. D. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201–213. · Zbl 1049.90004
[7] D. M. Gay, Computing optimal locally constrained steps, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 186–197. · Zbl 0467.65027
[8] G. H. Golub and U. von Matt, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561–580. · Zbl 0745.65029
[9] N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint, Solving the trust-region subproblem using the Lanczos method, SIAM J. Optim., 9 (1999), pp. 504–525. · Zbl 1047.90510
[10] N. I. M. Gould, D. Orban, and P. L. Toint, GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization, ACM Trans. Math. Software, 29 (2004), pp. 353–372. · Zbl 1068.90525
[11] N. I. M. Gould, D. P. Robinson, and H. S. Thorne, On solving trust-region and other regularised subproblems in optimization, Math. Program. Comput., 2 (2010), pp. 21–57. · Zbl 1193.65098
[12] W. W. Hager, Minimizing a quadratic over a sphere, SIAM J. Optim., 12 (2001), pp. 188–208. · Zbl 1058.90045
[13] W. W. Hager and Y. Krylyuk, Graph partitioning and continuous quadratic programming, SIAM J. Discrete Math., 12 (1999), pp. 500–523. · Zbl 0972.90086
[14] R.-C. Li, Vandermonde matrices with Chebyshev nodes, Linear Algebra Appl., 428 (2007), pp. 1803–1832. · Zbl 1140.65025
[15] R.-C. Li, On Meinardus’ examples for the conjugate gradient method, Math. Comp., 77 (2008), pp. 335–352. · Zbl 1128.65027
[16] R.-C. Li, Sharpness in rates of convergence for symmetric Lanczos method, Math. Comp., 79 (2010), pp. 419–435. · Zbl 1206.65132
[17] R.-C. Li and L.-H. Zhang, Convergence of block Lanczos method for eigenvalue clusters, Numer. Math., 131 (2015), pp. 83–113. · Zbl 1334.65073
[18] L. Lukšan, C. Matonoha, and J. Vlček, On Lagrange multipliers of trust-region subproblems, BIT, 48 (2008), pp. 763–768. · Zbl 1163.90027
[19] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, Berlin, 1967. · Zbl 0152.15202
[20] J. Moré and D. Sorensen, Computing a trust region step, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 553–572. · Zbl 0551.65042
[21] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006. · Zbl 1104.65059
[22] B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Appl. Math., SIAM, Philadelphia, 1998. · Zbl 0885.65039
[23] R. Rendl and H. Wolkowicz, A semidefinite framework for trust region subproblems with applications to large scale minimization, Math. Program., 77 (1997), pp. 273–299. · Zbl 0888.90137
[24] M. Rojas, S. A. Santos, and D. C. Sorensen, A new matrix-free algorithm for the large-scale trust-region subproblem, SIAM J. Optim., 11 (2000), pp. 611–646. · Zbl 0994.65067
[25] M. Rojas, S. A. Santos, and D. C. Sorensen, Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization, ACM Trans. Math. Software, 34 (2008), pp. 1–28. · Zbl 1291.65177
[26] M. Rojas and D. C. Sorensen, A trust-region approach to the regularization of large-scale discrete forms of ill-posed problems, SIAM J. Sci. Comput., 23 (2002), pp. 1843–1861. · Zbl 1006.86004
[27] Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal., 15 (1980), pp. 687–706. · Zbl 0456.65016
[28] Y. Saad, Iterative Methods for Linear Systems, PWS, Boston, 1996. · Zbl 1031.65047
[29] D. C. Sorensen, Newton’s method with a model trust region modification, SIAM J. Numer. Anal., 19 (1982), pp. 409–426. · Zbl 0483.65039
[30] D. C. Sorensen, Minimization of a large-scale quadratic function subject to a spherical constraint, SIAM J. Optim., 7 (1997), pp. 141–161. · Zbl 0878.65044
[31] T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), pp. 626–637. · Zbl 0518.65042
[32] A. Tarantola, Inverse Problem Theory, Elsevier, Amsterdam, The Netherlands, 1987. · Zbl 0875.65001
[33] A. N. Tikhonov, Regularization of incorrectly posed problems, Soviet Math., 4 (1963), pp. 1624–1627. · Zbl 0183.11601
[34] P. L. Toint, Towards an efficient sparsity exploiting Newton method for minimization, in Sparse Matrices and Their Uses, I. Duff, ed., Academic Press, London, 1981, pp. 57–88. · Zbl 0463.65045
[35] Y. Yuan, On the truncated conjugate gradient method, Math. Program., 87 (2000), pp. 561–573. · Zbl 0955.65039
[36] L.-H. Zhang, W. H. Yang, C. Shen, and J. Feng, Error Bounds of the Lanczos Approach for the Trust-Region Subproblem, Technical report, 2015, . · Zbl 1392.90088
[37] Y. Zhou and R.-C. Li, Bounding the spectrum of large Hermitian matrices, Linear Algebra Appl., 435 (2011), pp. 480–493. · Zbl 1221.15022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.