Inertial variable metric techniques for the inexact forward-backward algorithm

One of the most popular approaches for the minimization of a convex functional given by the sum of a differentiable term and a nondifferentiable one is the forward-backward method with extrapolation. The main reason making this method very appealing for a wide range of applications is that it achieves a O(1/k2) convergence rate in the objective function values, which is optimal for a first order method. Recent contributions on this topic are related to the convergence of the iterates to a minimizer and the possibility of adopting a variable metric in the proximal step. Moreover, it has been also proved that the objective function convergence rate is actually o(1/k2). However, these results are obtained under the assumption that the minimization subproblem involved in the backward step is computed exactly, which is clearly not realistic in a variety of relevant applications. In this paper, we analyze the convergence properties when both variable metric and inexact computation of the backward step are allowed. To do this, we adopt a suitable inexactness criterion and we devise implementable conditions on both the accuracy of the inexact backward step computation and the variable metric selection, so that the o(1/k2) rate and the convergence of the iterates are preserved. The effectiveness of the proposed approach is also validated with a numerical experience showing the effects of the combination of inexactness with variable metric techniques.