A Nested Lanczos Method for the Trust-Region Subproblem

The trust-region subproblem (TRS) minimizes a quadratic $f({$s$})={$s$}^{{ T}}H{$s$}/2+ {$s$}^{{T}}{$g$}$ over the ellipsoidal constraint $\|{$s$}\|_M\le \Delta$ for a symmetric and positive definite matrix $M$. For a large scale TRS, a Lanczos-type approach, namely, the generalized Lanczos trust-region (GLTR) method was introduced by Gould, Lucidi, Roma, and Toint [SIAM J. Optim., 9 (1999), pp. 504--525], and extends nicely the classical Lanczos method for the eigenvalue problem to TRS. Basically, GLTR attempts to obtain a feasible approximation in the Krylov subspace $\mathcal{K}_k(M^{-1}H,M^{-1}{$g$})$ in an efficient way. For an accurate approximation, the dimension $k$ of $\mathcal{K}_k(M^{-1}H,M^{-1}{$g$})$ is usually modest for a well-conditioned TRS, but can be large for ill-conditioned problems. This causes numerical difficulties in the computational costs, memory requirements, and numerical stability. This paper introduces an efficient nested restarting strategy for GLTR and resolves these numer...