Overrelaxation of the fast iterative shrinkage/thresholding algorithm for fast signal recovery

In this paper, we propose an overrelaxation scheme of FISTA and its monotone variant MFISTA. FISTA and MFISTA minimize the sum of a smooth and a nonsmooth convex function by iteratively applying a forward-backward splitting step. This problem setting arises naturally in the context of signal and image processing. The stepsize in the forward-backward splitting step of these methods is restricted to a constant value chosen from an interval that is determined by the Lipschitz constant of the gradient of the smooth function. The proposed scheme admits a variable stepsize that can be chosen from a larger interval than FISTA/MFISTA while keeping the same convergence rate as FISTA/MFISTA. A numerical example demonstrates the effectiveness of the proposed relaxations by showing that the proposed scheme outperforms the original FISTA and MFISTA in terms of speed of convergence.