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

Abstract

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. The generalized Lanczos trust-region (GLTR) method is a well-known Lanczos type approach for solving a large-scale TRS. The method projects the original large-scale TRS onto a sequence of lower dimensional Krylov subspaces, whose orthonormal bases are generated by the symmetric Lanczos process, and computes approximate solutions from the underlying subspaces. There have been some a priori bounds available for the errors of the approximate solutions and approximate objective values obtained by the GLTR method, but no a priori bound exists on the errors of the approximate Lagrangian multipliers and the residual norms of approximate solutions obtained by the GLTR method. In this paper, a general convergence theory of the GLTR method is established for the TRS in the easy case, showing that the a priori bounds for these four quantities are closely interrelated and the one for the computable residual norm is of crucial importance in both theory and practice as it can predict the sizes of other three uncomputable errors reliably. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the four quantities accurately.