  In this note, we consider a special instance of the scaled, inexact and
adaptive generalised Fast Iterative Soft-Thresholding Algorithm (SAGE-FISTA)
recently proposed in (Rebegoldi, Calatroni, '21) for the efficient solution of
strongly convex composite optimisation problems. In particular, we address here
the sole (non-strongly) convex optimisation scenario, which is frequently
encountered in many imaging applications. The proposed inexact S-FISTA
algorithm shows analogies to the variable metric and inexact version of FISTA
studied in (Bonettini, Rebegoldi, Ruggiero, '19), the main difference being the
use of an adaptive (non-monotone) backtracking strategy allowing for the
automatic adjustment of the algorithmic step-size along the iterations (see
(Scheinberg, Goldfarb, Bai, '14, Calatroni, Chambolle, '19)). A quadratic
convergence result in function values depending on the backtracking parameters
and the upper and lower bounds on the spectrum of the variable metric operators
is given. Experimental results on TV image deblurring problems with Poisson
noise are then reported for numerical validation, showing improved
computational efficiency and precision.
