NDLI: On the Generalized Lanczos Trust-Region Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shen, Chungen Li, Ren-Cang Zhang, Lei-Hong
Copyright Year	2017
Abstract	The so-called trust-region subproblem gets its name in the trust-region method in optimization and also plays a vital role in various other applications. Several numerical algorithms have been proposed in the literature for solving small-to-medium size dense problems as well as for large-scale sparse problems. The generalized Lanczos trust-region (GLTR) method proposed by [N. I. M. Gould, S. Lucidi, M. Roma and P. L. Toint, SIAM J. Optim., 9 (1999), pp. 561--580] is a natural extension of the classical Lanczos method for the linear system to the trust-region subproblem. In this paper, we first analyze the convergence of GLTR to reveal its convergence behavior in theory and then propose new stopping criteria that can be integrated into GLTR for better numerical performance. Specifically, we develop a priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution and argue that these bounds can be efficiently estimated numerically and serve as stopping criteria for iterative methods such as GLTR. Two sets of numerical tests are presented. In the first set, we demonstrate the sharpness of the upper bounds, and for the second set, we integrate the upper bound estimate into the Fortran routine GLTR in the library GALAHAD as new stopping criteria and test the trust-region solver TRU on the problem collection CUTEr. The numerical results show that, with the new stopping criteria in \textttGLTR, the overall performance of TRU can be improved considerably.
Sponsorship	Shanghai University of Finance and Economics. National Natural Science Foundation of China. National Science Foundation
Starting Page	2110
Ending Page	2142
Page Count	33
File Format	PDF
ISSN	10526234
DOI	10.1137/16M1095056
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-09-28
Access Restriction	Subscribed
Subject Keyword	trust-region method convergence Iterative methods for linear systems stopping criterion Eigenvalues, eigenvectors conjugate gradient method Quadratic programming Matrix norms, conditioning, scaling trust-region subproblem Lanczos method Large-scale problems
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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