In a Hilbert space $\mathcal{H}$ H , we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $ A = ∇ f + B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as $t\to +\infty $ t → + ∞ of the generated trajectories towards the zeros of $\nabla f +B$ ∇ f + B . The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. 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 and differ from the classical ones due to the presence of the nonpotential term.

In a Hilbert space setting, we study a class of first-order algorithms which aim to solve structured monotone equations governed by sums of potential and nonpotential operators. Precisely, we are looking for the zeros of an operator A = ∇f + B where ∇f is the gradient of a differentiable convex function f , and B is a nonpotential monotone and cocoercive operator. Our study is based on the inertial autonomous dynamic previously studied by the authors to solve this type of problem, and which involves dampings which are respectively controlled by the Hessian of f , and by a Newton-type correction term attached to B. These geometric dampings attenuate the oscillations which occur with the inertial methods with viscous damping. Using Lyapunov analysis, we study the convergence properties of the proximalgradient algorithms obtained by temporal discretization of this dynamic. These results open the door to the design of first-order accelerated algorithms in numerical optimization taking into account the specific properties of potential and nonpotential terms.

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