A Doubly Nonlinear Evolution System with Threshold Effects Associated with Dry Friction

Abstract

In this paper, we investigate the asymptotic behavior of inertial dynamics with dry friction within the context of a Hilbert framework for convex differentiable optimization. Our study focuses on a doubly nonlinear first-order evolution inclusion that encompasses two potentials. In our analysis, we specifically focus on two main components: the differentiable function f that needs to be minimized, which influences the system's state through its gradient, and the nonsmooth dry friction potential denoted as ϕ = r • . It's important to note that the dry friction term acts on a linear combination of the velocity vector and the gradient of f . Consequently, any stationary point in our system corresponds to a critical point of f , unlike the case where only the velocity vector is involved in the dry friction term, resulting in an approximate critical point of f . To emphasize the crucial role of ∇f (x), we also explore the dual formulation of this dynamic, which possesses a Riemannian gradient structure. To address these dynamics, we employ the recently developed generic acceleration approach by Attouch, Bot, and Nguyen. This approach involves the time scaling of a continuous first-order differential equation, followed by the application of the method of averaging. By applying this methodology, we derive fast convergence results for second-order time-evolution systems with dry friction, asymptotically vanishing viscous damping, and implicit Hessian-driven damping.