Multi-step inertial Krasnosel'skii-Mann iteration with new inertial parameters arrays

Recently, the authors (Dong et al. in J Global Optim 73(4):801-824, 2019) introduced the multi-step inertial Krasnosel'skii-Mann iteration, where the inertial parameters involve the iterative sequence. Therefore, one has to compute the inertial parameters per iteration. The aim of this article is to present two kinds of inertial parameter arrays which do not depend on the iterative sequence. We first introduce a general Krasnosel'skii-Mann iteration on the affine hull of orbits, based on which one inertial parameter array is presented. Second, we investigate the other inertial parameter array by introducing a modified Krasnosel'skii-Mann iteration. The convergence of the modified Krasnosel'skii-Mann iteration is shown using an exhaustive convergence analysis and the running-average iteration-complexity bound is provided. Finally, we give two numerical examples to illustrate that the multi-step inertial Krasnosel'skii-Mann iteration with inertial parameters proposed in this article behaves better than that with inertial parameters given in [10].