Scaled, Inexact, and Adaptive Generalized FISTA for Strongly Convex Optimization

We consider a variable metric and inexact version of the fast iterative soft-thresholding algorithm (FISTA) type algorithm considered in [L. Calatroni and A. Chambolle, SIAM J. Optim., 29 (2019), pp. 1772--1798; A. Chambolle and T. Pock, Acta Numer., 25 (2016), pp. 161--319] for the minimization of the sum of two (possibly strongly) convex functions. The proposed algorithm is combined with an adaptive (nonmonotone) 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 is common in applications where signal-dependent Poisson noise is assumed in the data.