A Continuous-Time Perspective on Optimal Methods for Monotone Equation Problems

Abstract

We study rescaled gradient dynamical systems in a Hilbert space H, where the implicit discretization in a finite-dimensional Euclidean space leads to high-order methods for solving monotone equations (MEs). Our framework generalizes the celebrated dual extrapolation method [Nesterov, 2007] from first order to high order via appeal to the regularization toolbox of optimization theory [Nesterov, 2021a,b]. We establish the existence and uniqueness of a global solution and analyze the convergence properties of solution trajectories. We also present discrete-time counterparts of our high-order continuous-time methods, and we show that the p th -order method achieves an ergodic rate of O(k −(p+1)/2 ) in terms of a restricted merit function and a pointwise rate of O(k −p/2 ) in terms of a residue function. Under regularity conditions, the restarted version of p th -order methods achieves local convergence with the order p ≥ 2.