Analysis of Krylov Subspace Solutions of Regularized Nonconvex Quadratic Problems

We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We prove error bounds of the form $1\mathrm{/}{t}^2$ and $e^{-4t\mathrm{/}\sqrt{\kappa }}$, where $\kappa$ is a condition number for the problem, and $t$ is the Krylov subspace order (number of Lanczos iterations). We also provide lower bounds showing that our analysis is sharp.