  We consider a variable metric and inexact version of the FISTA-type algorithm
considered in (Chambolle, Pock, 2016, Calatroni, Chambolle, 2019) for the
minimization of the sum of two (possibly strongly) convex functions. The
proposed algorithm is combined with an adaptive (non-monotone) backtracking
strategy, which allows for the adjustment of the algorithmic step-size along
the iterations in order to improve the convergence speed. We prove a linear
convergence result for the function values, which depends on both the strong
convexity moduli of the two functions and the upper and lower bounds on the
spectrum of the variable metric operators. We validate the proposed algorithm,
named Scaled Adaptive GEneralized FISTA (SAGE-FISTA), on exemplar image
denoising and deblurring problems where edge-preserving Total Variation (TV)
regularization is combined with Kullback-Leibler-type fidelity terms, as it is
common in applications where signal-dependent Poisson noise is assumed in the
data.
