Shifted L-BFGS Systems

We investigate fast direct methods for solving systems of the form (B + G)x = y, where B is a limited-memory BFGS 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 and stable manner via a recursion that requies only vector inner products. We consider various shift matrices G and demonstrate the effectiveness of the recursion methods in numerical experiments.