A ContinuousTime Perspective on Optimal Methods for Monotone Equation Problems
Abstract
We study \textit{rescaled gradient dynamical systems} in a Hilbert space $\mathcal{H}$, where implicit discretization in a finitedimensional Euclidean space leads to highorder methods for solving monotone equations (MEs). Our framework can be interpreted as a natural generalization of celebrated dual extrapolation method~\citep{Nesterov2007Dual} from first order to high order via appeal to the regularization toolbox of optimization theory~\citep{Nesterov2021Implementable, Nesterov2021Inexact}. More specifically, we establish the existence and uniqueness of a global solution and analyze the convergence properties of solution trajectories. We also present discretetime counterparts of our highorder continuoustime 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 $p \geq 2$. Notably, our methods are \textit{optimal} since they have matched the lower bound established for solving the monotone equation problems under a standard linear span assumption~\citep{Lin2022Perseus}.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.04770
 Bibcode:
 2022arXiv220604770L
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 35 Pages