A nested primal–dual FISTA-like scheme for composite convex optimization problems

We propose a nested primal–dual algorithm with extrapolation on the primal variable suited for minimizing the sum of two convex functions, one of which is continuously differentiable. The proposed algorithm can be interpreted as an inexact inertial forward–backward algorithm equipped with a prefixed number of inner primal–dual iterations for the proximal evaluation and a “warm–start” strategy for starting the inner loop, and generalizes several nested primal–dual algorithms already available in the literature. By appropriately choosing the inertial parameters, we prove the convergence of the iterates to a saddle point of the problem, and provide an O(1/n) convergence rate on the primal–dual gap evaluated at the corresponding ergodic sequences. Numerical experiments on some image restoration problems show that the combination of the “warm–start” strategy with an appropriate choice of the inertial parameters is strictly required in order to guarantee the convergence to the real minimum point of the objective function.