Accelerated forward–backward algorithms for structured monotone inclusions

In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of the classical forward–backward method is considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term, along with a preconditioning process. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates \((x_n)\) $(x_n)$ , with worst-case rates of \( o(n^{-1})\) $o(n^{-1})$ in terms of both the discrete velocity and the fixed point residual, instead of the rates of \(\mathcal {O}(n^{-1/2})\) $\mathcal{O}(n^{-1/2})$ classically established for related algorithms. Our procedure can be also adapted to more general monotone inclusions. In particular, we propose a fast primal-dual algorithmic solution to some class of convex-concave saddle point problems. In addition, we provide a well-adapted framework for solving this class of problems by means of standard proximal-like algorithms dedicated to structured monotone inclusions. Numerical experiments are also performed so as to enlighten the efficiency of the proposed strategy.