  We propose the nonlinear restricted additive Schwarz (RAS) preconditioning
strategy to improve the convergence speed of limited memory quasi-Newton (QN)
methods. We consider both "left-preconditioning" and "right-preconditioning"
strategies. As the application of the nonlinear preconditioning changes the
standard gradients and Hessians to their preconditioned counterparts, the
standard secant pairs cannot be used to approximate the preconditioned
Hessians. We discuss how to construct the secant pairs in the preconditioned QN
framework. Finally, we demonstrate the robustness and efficiency of the
preconditioned QN methods using numerical experiments.
