Convergence rate of inertial forward–backward algorithms based on the local error bound condition

Abstract

Abstract The ‘inertial forward–backward algorithm’ (IFB) is a powerful algorithm for solving a class of convex non-smooth minimization problems, IFB relies on an inertial parameter $\gamma _{k}$ whose tuning is crucial for achieving accelerated convergence speeds as compared to the classical forward–backward algorithm. Under the local error bound condition, it is known that IFB converges R-linearly as soon as the inertial parameter satisfies ${\sup _{k}}{\gamma _{k}} \leqslant \tilde{\gamma } <1.$ On the contrary, we are not aware of any convergence result for the case ${\sup _{k}}{\gamma _{k}} = 1.$ In this paper, we consider six different choices of inertial parameters satisfying this last condition, and show convergence of the corresponding IFB algorithms under the local error bound condition. Finally, we propose a class of inertial forward–backward algorithm with an adaptive modification (IFB_AdapM) and show that it enjoys the same convergence results.