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

We study rescaled gradient dynamical systems in a Hilbert space H , where implicit discretization in a finite-dimensional Euclidean space leads to high-order methods for solving monotone equations (MEs). Our framework can be interpreted as a natural generalization of celebrated dual extrapolation method [Nesterov, 2007] from first order to high order via appeal to the regularization toolbox of optimization theory [Nesterov, 2021a,b]. More specifically, 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. Notably, our methods are optimal since they have matched the lower bound established for solving the monotone equation problems under a standard linear span assumption [Lin and Jordan, 2022].