NEW INERTIAL METHOD FOR NONEXPANSIVE MAPPINGS
Abstract
There have been increasing interests in studying inertial Krasnoselskii-Mann iterations due to the presence of inertial extrapolation step which improves the rate of convergence of Krasnoselskii-Mann iterations. These results analyzed the convergence properties of inertial Krasnoselskii-Mann iterations and demonstrated their performance numerically on someimaging and data analysis problems. It is discovered that these
proposed inertial Krasnoselskii-Mann iterations assumed some
stringent conditions on the inertial factor which make the implementations difficult in some numerical examples. In this present paper, we provide a new inertial Krasnoselskii-Mann iteration, prove its weak convergence and the corresponding rate of convergence under some suitable conditions.
References
F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal. 9 3-11, (2001).
H. Attouch, X. Goudon and P. Redont, The heavy ball with friction. I. The continuous dynamical system, Commun. Contemp. Math. 2 (1) 1-34, (2000).
H. Attouch and M.O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Differential Equations 179 (1) 278-310, (2002).
H. Attouch, J. Peypouquet and P. Redont, A dynamical approach to an inertial forward-backward algorithm for convex minimization, SIAM J. Optim. 24 232-256, (2014).
H. Attouch and J. Peypouquet, The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than $frac{1}{k^2}$ , SIAM J. Optim. 26 1824-1834, (2016).
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York (2011).
