Convergence of Inertial Dynamics Driven by Sums of Potential and Nonpotential Operators with Implicit Newton-Like Damping

We analyze the convergence properties when the time t tends to infinity of the trajectories generated by damped inertial dynamics which are driven by the sum of potential and nonpotential operators. Precisely, we seek to reach asymptotically the zeros of a maximally monotone operator which is the sum of a potential operator (the gradient of a continuously differentiable convex function) and of a monotone and cocoercive nonpotential operator. As an original feature, in addition to the viscous friction, the dynamic involves implicit Newton-type damping. This contrasts with the authors’ previous study where explicit Newton-type damping was considered, which, for the potential term corresponds to Hessian-driven damping. We show the weak convergence, as time tends to infinity, of the generated trajectories toward the zeros of the sum of the potential and nonpotential operators. Our results are based on Lyapunov analysis and appropriate setting of the damping parameters. The introduction of geometric dampings allows to control and attenuate the oscillations known for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a system involving only first-order derivative in time and space allows us to extend the convergence analysis to nonsmooth convex potentials. The main part of our study concerns the autonomous case with positive fixed parameters. We complete it with some first results concerning the nonautonomous case, and which are based on a recent acceleration method using time scaling and averaging. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original due to the presence of the nonpotential term.