Newton-Type Inertial Algorithms for Solving Monotone Equations Governed by Sums of Potential and Nonpotential Operators

Abstract

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.