Fast convergence of generalized forward-backward algorithms for structured monotone inclusions

In this paper, we develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical forward-backward method for two general operators is first considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $(x_n)$, with worst-case rates of $o(n^{-2})$ in terms of both the discrete velocity and the fixed point residual, instead of the classical rates of $\mathcal{O}(n^{-1})$ established so far for related algorithms. Our procedure is then adapted to more general monotone inclusions and a fast primal-dual algorithm is proposed for solving convex-concave saddle point problems.