Accelerated dynamics with dry friction via time scaling and averaging of doubly nonlinear evolution equations

Abstract

In a Hilbert framework, for convex differentiable optimization, we analyze the long-time behavior of inertial dynamics with dry friction. In classical approaches based on asymptotically vanishing viscous damping (in accordance with Nesterov’s method), the results are expressed in terms of rapid convergence of the values of the function to its minimum value. On the other hand, dry friction induces convergence towards an approximate minimizer, typically the system stops at x when a given threshold ‖∇f(x)‖≤r is satisfied. We obtain rapid convergence results in this direction. In our approach, we start from a doubly nonlinear first-order evolution equation involving two potentials: one is the differentiable function f to be minimized, which acts on the state of the system via its gradient, and the other is the nonsmooth potential dry friction φ(x)=r‖x‖ which acts on the velocity vector via its sub-differential. To highlight the central role played by ∇f(x), we also deal with the dual formulation of these dynamics, which have a Riemannian gradient structure. We then rely on the general acceleration method recently developed by Attouch, Boţ and Nguyen, which consists in applying the method of time scaling and then averaging to a continuous differential equation of the first order in time. We thus obtain fast convergence results for second order time-evolution systems involving dry friction, asymptotically vanishing viscous damping, and Hessian-driven damping in the implicit form.