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Solving the trust-region subproblem (TRS) plays a key role in numericaloptimization and many other applications. Based on a fundamental result thatthe solution of TRS of size $n$ is mathematically equivalent to finding therightmost eigenpair of a certain matrix pair of size $2n$, eigenvalue-basedmethods are promising due to their simplicity. For $n$ large, the implicitlyrestarted Arnoldi (IRA) and refined Arnoldi (IRRA) algorithms are well suitedfor this eigenproblem. For a reasonable comparison of overall efficiency of thealgorithms for solving TRS directly and eigenvalue-based algorithms, a vitalpremise is that the two kinds of algorithms must compute the approximatesolutions of TRS with (almost) the same accuracy, but such premise has beenignored in the literature. To this end, we establish close relationshipsbetween the two kinds of residual norms, so that, given a stopping tolerancefor IRA and IRRA, we are able to determine a reliable one that GLTR should useso as to ensure that GLTR and IRA, IRRA deliver the converged approximatesolutions with similar accuracy. We also make a convergence analysis on theresidual norms by the Generalized Lanczos Trust-Region (GLTR) algorithm forsolving TRS directly, the Arnoldi method and the refined Arnoldi method for theequivalent eigenproblem. A number of numerical experiments are reported toillustrate that IRA and IRRA are competitive with GLTR and IRRA outperformsIRA.

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