  In [19], a general, inexact, efficient proximal quasi-Newton algorithm for
composite optimization problems has been proposed and a sublinear global
convergence rate has been established. In this paper, we analyze the
convergence properties of this method, both in the exact and inexact setting,
in the case when the objective function is strongly convex. We also investigate
a practical variant of this method by establishing a simple stopping criterion
for the subproblem optimization. Furthermore, we consider an accelerated
variant, based on FISTA [1], to the proximal quasi-Newton algorithm. A similar
accelerated method has been considered in [7], where the convergence rate
analysis relies on very strong impractical assumptions. We present a modified
analysis while relaxing these assumptions and perform a practical comparison of
the accelerated proximal quasi- Newton algorithm and the regular one. Our
analysis and computational results show that acceleration may not bring any
benefit in the quasi-Newton setting.
