Paper Info
Title
The Convergence Of The Generalized Lanczos Trust-Region Method For The Trust-Region Subproblem
Abstract
Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. The generalized Lanczos trust-region (GLTR) method is a well-known Lanczos type approach for solving a large-scale TRS. The method projects the original large-scale TRS onto a sequence of lower dimensional Krylov subspaces, whose orthonormal bases are generated by the symmetric Lanczos process, and computes approximate solutions from the underlying subspaces. There have been some a priori bounds available for the errors of the approximate solutions and approximate objective values obtained by the GLTR method, but no a priori bound exists on the errors of the approximate Lagrangian multipliers and the residual norms of approximate solutions obtained by the GLTR method. In this paper, a general convergence theory of the GLTR method is established for the TRS in the easy case, showing that the a priori bounds for these four quantities are closely interrelated and the one for the computable residual norm is of crucial importance in both theory and practice as it can predict the sizes of other three uncomputable errors reliably. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the four quantities accurately.
Year
DOI
Venue
2021
10.1137/19M1279691
SIAM JOURNAL ON OPTIMIZATION
Keywords
DocType
Volume
trust-region subproblem, GLTR method, a priori bound, easy case, hard case, Chebyshev polynomial, eigenvalue problem, symmetric Lanczos process
Journal
31
Issue
ISSN
Citations 
1
1052-6234
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Zhongxiao Jia112118.57
Fa Wang2558.89