Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization

In a Hilbert space, we analyze the convergence properties of a general class of inertial forward–backward algorithms in the presence of perturbations, approximations, errors. These splitting algorithms aim to solve, by rapid methods, structured convex minimization problems. The function to be minimized is the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous convex function. The algorithms involve a general sequence of positive extrapolation coefficients that reflect the inertial effect and a sequence in the Hilbert space that takes into account the presence of perturbations. We obtain convergence rates for values and convergence of the iterates under conditions involving the extrapolation and perturbation sequences jointly. This extends the recent work of Attouch–Cabot which was devoted to the unperturbed case. Next, we consider the introduction into the algorithms of a Tikhonov regularization term with vanishing coefficient. In this case, when the regularization coefficient does not tend too rapidly to zero, we obtain strong ergodic convergence of the iterates to the minimum norm solution. Taking a general sequence of extrapolation coefficients makes it possible to cover a wide range of accelerated methods. In this way, we show in a unifying way the robustness of these algorithms.