We design an inexact, scaled and generalised Fast Iterative Soft-Thresholding Algorithm (FISTA) for minimising the sum of two (possibly strongly) convex functions. Inexactness is explicitly taken into account for describing situations where proximal operators cannot be evaluated in closed form. The use of data-dependent scaling allows to incorporate Newton-type information along the optimisation via variable-metric updates. Finally, non-monotone backtracking is used to improve convergence speed. Linear convergence for the function values is proved with rates depending on backtracking/scaling and strong convexity parameters. The performance of the proposed algorithm, named SAGE-FISTA, is validated on exemplar imaging problems where sparsity-promoting (l_1, TV) regularisation is combined with popular data-fidelity terms. Numerical results show improved performance and practical efficiency under limited computational budget.
