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

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