Shifted L-BFGS systems
Jennifer B. Erway, Vibhor Jain, Roummel F. Marcia · Optimization methods & software · 2014
We investigate fast direct methods for solving systems of the form (B+G)x=y, where B is a limited-memory Broyden-Fletcher-Goldfarb-Shanno matrix and G is a symmetric positive-definite matrix. These systems, which we refer to as shifted L-BFGS systems, arise in several settings, including trust-region methods and preconditioning techniques for interior-point methods. We show that under mild assumptions, the system (B+G)x=y can be solved in an efficient manner that mitigates instability via a recursion that requires only vector inner products. We consider various shift matrices G and demonstrate the effectiveness and accuracy of the recursion method in numerical experiments.