A comparison of eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for Solving the trust-region subproblem

02/19/2021
by   Zhongxiao Jia, et al.
0

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. Based on a fundamental result that the solution of TRS of size n is mathematically equivalent to finding the rightmost eigenpair of a certain matrix pair of size 2n, eigenvalue-based methods are promising due to their simplicity. For n large, the implicitly restarted Arnoldi (IRA) and refined Arnoldi (IRRA) algorithms are well suited for this eigenproblem. For a reasonable comparison of overall efficiency of the algorithms for solving TRS directly and eigenvalue-based algorithms, a vital premise is that the two kinds of algorithms must compute the approximate solutions of TRS with (almost) the same accuracy, but such premise has been ignored in the literature. To this end, we establish close relationships between the two kinds of residual norms, so that, given a stopping tolerance for IRA and IRRA, we are able to determine a reliable one that GLTR should use so as to ensure that GLTR and IRA, IRRA deliver the converged approximate solutions with similar accuracy. We also make a convergence analysis on the residual norms by the Generalized Lanczos Trust-Region (GLTR) algorithm for solving TRS directly, the Arnoldi method and the refined Arnoldi method for the equivalent eigenproblem. A number of numerical experiments are reported to illustrate that IRA and IRRA are competitive with GLTR and IRRA outperforms IRA.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

08/06/2019

The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem

Solving the trust-region subproblem (TRS) plays a key role in numerical ...
07/20/2019

The Generalized Trust Region Subproblem: solution complexity and convex hull results

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing...
12/27/2021

Implicit regularity and linear convergence rates for the generalized trust-region subproblem

In this paper we develop efficient first-order algorithms for the genera...
08/11/2020

On inner iterations of the joint bidiagonalization based algorithms for solving large scale linear discrete ill-posed problems

The joint bidiagonalization process of a large matrix pair A,L can be us...
10/22/2020

One-shot Distributed Algorithm for Generalized Eigenvalue Problem

Nowadays, more and more datasets are stored in a distributed way for the...
03/17/2021

An inexact Douglas-Rachford splitting method for solving absolute value equations

The last two decades witnessed the increasing of the interests on the ab...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.