Abstract
The ‘heavy ball with friction’ dynamical system x + γx + ∇f(x)=0 is a nonlinear oscillator with damping (γ>0). It has been recently proved that when H is a real Hilbert space and f: H→R is a differentiable convex function whose minimal value is achieved, then each solution trajectory t→x(t) of this system weakly converges towards a solution of ∇f(x)=0. We prove a similar result in the discrete setting for a general maximal monotone operator A by considering the following iterative method: x k+1−x k−α k (x k−x k−1)+λ k A(x k+1)∋0, giving conditions on the parameters λ k and α k in order to ensure weak convergence toward a solution of 0∈A(x) and extending classical convergence results concerning the standard proximal method.
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Alvarez, F., Attouch, H. An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping. Set-Valued Analysis 9, 3–11 (2001). https://doi.org/10.1023/A:1011253113155
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DOI: https://doi.org/10.1023/A:1011253113155