Simultaneous iterative solutions for the trust-region and minimum eigenvalue subproblem

Given the inability to foresee all possible scenarios, it is justified to desire an efficient trust-region subproblem solver capable of delivering any desired level of accuracy on demand; that is, the accuracy obtainable for a given trust-region subproblem should not be partially dependent on the problem itself. Current state-of-the-art iterative eigensolvers all fall into the class of restarted Lanczos methods; whereas, current iterative trust-region solvers at best reduce to unrestarted Lanczos methods; which in this context are well known to be numerically unstable with impractical memory requirements. In this paper, we present the first iterative trust region subproblem solver that at its core contains a robust and practical eigensolver. Our solver leverages the recently announced work of Stathopoulos and Orginos which has not been noticed by the optimization community and can be utilized because, unlike other restarted Lanczos methods, its restarts do not necessarily modify the current Lanczos sequence generated by Conjugate Gradient methods (CG). This innovated strategy can be utilized in the context of TR solvers as well. Moreover, our TR subproblem solver adds negligible computational overhead compared to existing iterative TR approaches.