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We address the problem of finding the zeros of the sum of a maximallymonotone operator and a cocoercive operator. Our approach introduces amodification to the forward-backward method by integrating an inertial/momentumterm alongside a correction term. We demonstrate that the sequence ofiterations thus generated converges weakly towards a solution for the monotoneinclusion problem. Furthermore, our analysis reveals an outstanding attributeof our algorithm: it displays rates of convergence of the order $o(1/k)$ forthe discrete velocity and the tangent residual approaching zero. These ratesfor tangent residuals can be extended to fixed-point residuals frequentlydiscussed in the existing literature. Specifically, when applied to minimize anonsmooth convex function subject to linear constraints, our method evolvesinto a primal-dual full splitting algorithm. Notably, alongside the convergenceof iterates, this algorithm possesses a remarkable characteristic ofnonergodic/last iterate $o(1/k)$ convergence rates for both the function valueand the feasibility measure. Our algorithm showcases the most advancedconvergence and convergence rate outcomes among primal-dual full splittingalgorithms when minimizing nonsmooth convex functions with linear constraints.

Fast Forward-Backward splitting for monotone inclusions with a convergence rate of the tangent residual of $o(1/k)$