An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping

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: xk+1−xk−αk(xk−xk−1)+λkA(xk+1)∋0, giving conditions on the parameters λk and αk in order to ensure weak convergence toward a solution of 0∈A(x) and extending classical convergence results concerning the standard proximal method.