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In [19], a general, inexact, efficient proximal quasi-Newton algorithm forcomposite optimization problems has been proposed and a sublinear globalconvergence rate has been established. In this paper, we analyze theconvergence properties of this method, both in the exact and inexact setting,in the case when the objective function is strongly convex. We also investigatea practical variant of this method by establishing a simple stopping criterionfor the subproblem optimization. Furthermore, we consider an acceleratedvariant, based on FISTA [1], to the proximal quasi-Newton algorithm. A similaraccelerated method has been considered in [7], where the convergence rateanalysis relies on very strong impractical assumptions. We present a modifiedanalysis while relaxing these assumptions and perform a practical comparison ofthe accelerated proximal quasi- Newton algorithm and the regular one. Ouranalysis and computational results show that acceleration may not bring anybenefit in the quasi-Newton setting.

Proximal Quasi-Newton Methods for Regularized Convex Optimization with Linear and Accelerated Sublinear Convergence Rates