Halpern Acceleration of the Inexact Proximal Point Method of Rockafellar (2511.10372v1)

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.