Frontiers of Mathematics in China >
Error bounds of Lanczos approach for trust-region subproblem
Received date: 06 Jan 2016
Accepted date: 06 Feb 2018
Published date: 28 Mar 2018
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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.
Leihong ZHANG , Weihong YANG , Chungen SHEN , Jiang FENG . Error bounds of Lanczos approach for trust-region subproblem[J]. Frontiers of Mathematics in China, 2018 , 13(2) : 459 -481 . DOI: 10.1007/s11464-018-0687-y
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