Halpern Acceleration of the Inexact Proximal Point Method of Rockafellar

TL;DR

This paper introduces a Halpern acceleration technique for the inexact proximal point method, achieving faster convergence rates for solving monotone inclusion problems and extending to constrained convex optimization.

Contribution

It develops the HiPPM algorithm with proven global convergence and improved convergence rates, extending the acceleration to augmented Lagrangian methods.

Findings

Achieves an $ ext{O}(1/k^2)$ convergence rate in fixed-point residuals.

Attains linear convergence under regularity conditions.

Numerical experiments confirm theoretical convergence improvements.

Abstract

This paper investigates a Halpern acceleration of the inexact proximal point method for solving maximal monotone inclusion problems in Hilbert spaces. The proposed Halpern inexact proximal point method (HiPPM) is shown to be globally convergent, and a unified framework is developed to analyze its worst-case convergence behavior. Under mild conditions on the inexactness tolerances, HiPPM achieves an $\mathcal{O}(1/k^{2})$ convergence rate in terms of the squared fixed-point residual. Moreover, under additional well-studied regularity conditions, the method attains a fast linear convergence rate. Building on this framework, we further extend the Halpern acceleration to the inexact augmented Lagrangian method for constrained convex optimization. In the spirit of Rockafellar's classical results, the resulting accelerated inexact augmented Lagrangian method inherits the convergence rate and iteration complexity guarantees of HiPPM. Numerical experiments are provided to support the theoretical findings.