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

TL;DR

This paper compares eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for solving the trust-region subproblem, establishing residual norm relationships and analyzing convergence to ensure fair efficiency comparisons.

Contribution

It introduces a method to compare algorithms fairly by matching solution accuracy and provides convergence analysis for these algorithms in solving TRS.

Findings

IRA and IRRA are competitive with GLTR.

IRRA outperforms IRA in efficiency.

Residual norm relationships enable fair algorithm comparisons.

Abstract

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.