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Halpern Acceleration of the Inexact Proximal Point Method of Rockafellar

Liwei Zhang, Fanli Zhuang, Ning Zhang

TL;DR

This work addresses solving maximal monotone inclusions in a real Hilbert space by introducing a Halpern-accelerated inexact proximal point framework (HiPPM). By integrating Halpern-type updates with inexact proximal steps, the authors prove global convergence and derive a unified rate analysis showing an $\mathcal{O}(1/k^{2})$ decay in the squared fixed-point residual under mild summability of the inexactness tolerances, plus potential linear convergence under strong monotonicity. They further extend the framework to constrained convex optimization via the augmented Lagrangian method, yielding accelerated inexact ALM guarantees that preserve convergence rates and complexity. Collectively, the results provide a cohesive theory linking accelerated proximal algorithms and their augmented Lagrangian extensions, with explicit rate bounds tied to tolerance sequences and problem regularity.

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 rate. Under mild summability conditions on the inexactness tolerances, HiPPM achieves an $\mathcal{O}(1/k^{2})$ rate in terms of the squared fixed-point residual. Furthermore, under additional mild condition, the method retains a fast linear convergence rate. Building upon this framework, we further extend the acceleration technique to constrained convex optimization through the augmented Lagrangian formulation. In analogy to Rockafellar's classical results, the resulting accelerated inexact augmented Lagrangian method inherits the convergence rate and complexity guarantees of HiPPM. The analysis thus provides a unified theoretical foundation for accelerated inexact proximal algorithms and their augmented Lagrangian extensions.

Halpern Acceleration of the Inexact Proximal Point Method of Rockafellar

TL;DR

This work addresses solving maximal monotone inclusions in a real Hilbert space by introducing a Halpern-accelerated inexact proximal point framework (HiPPM). By integrating Halpern-type updates with inexact proximal steps, the authors prove global convergence and derive a unified rate analysis showing an decay in the squared fixed-point residual under mild summability of the inexactness tolerances, plus potential linear convergence under strong monotonicity. They further extend the framework to constrained convex optimization via the augmented Lagrangian method, yielding accelerated inexact ALM guarantees that preserve convergence rates and complexity. Collectively, the results provide a cohesive theory linking accelerated proximal algorithms and their augmented Lagrangian extensions, with explicit rate bounds tied to tolerance sequences and problem regularity.

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 rate. Under mild summability conditions on the inexactness tolerances, HiPPM achieves an rate in terms of the squared fixed-point residual. Furthermore, under additional mild condition, the method retains a fast linear convergence rate. Building upon this framework, we further extend the acceleration technique to constrained convex optimization through the augmented Lagrangian formulation. In analogy to Rockafellar's classical results, the resulting accelerated inexact augmented Lagrangian method inherits the convergence rate and complexity guarantees of HiPPM. The analysis thus provides a unified theoretical foundation for accelerated inexact proximal algorithms and their augmented Lagrangian extensions.

Paper Structure

This paper contains 4 sections, 13 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Halpern acceleration of the inexact PPA
  3. The rate of convergence
  4. Accelerated inexact augmented Lagrange method

Key Result

Lemma 2.1

Suppose that for some $\tilde{z} \in {\cal H}$ and $\rho >0$ one has Then there exists at least one $z$ satisfying $0\in T(z)$. Moreover, this condition is both necessary and sufficient for the existence of a solution.

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 14 more