Limited-memory BFGS systems with diagonal updates

In this paper, we investigate a formula to solve systems of the form (B+σI)x=y, where B is a limited-memory BFGS quasi-Newton matrix and σ is a positive constant. These types of systems arise naturally in large-scale optimization such as trust-region methods as well as doubly-augmented Lagrangian methods. We show that provided a simple condition holds on B0 and σ, the system (B+σI)x=y can be solved via a recursion formula that requires only vector inner products. This formula has complexity M2n, where M is the number of L-BFGS updates and n≫M is the dimension of x.