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Stabilized Proximal Point Method via Trust Region Control

Hanmin Li, Kaja Gruntkowska, Peter Richtárik

Abstract

The Proximal Point Method (PPM) (Rockafellar, 1976) is a fundamental tool for nonsmooth convex optimization. However, its convergence is not linear under general convexity in the absence of strong convexity or other structural assumptions. To address this limitation, we study a trust-region stabilized proximal point scheme in which each proximal update is computed over a localized feasible region. We show that this simple stabilization enforces non-vanishing steps and yields a linear decrease in objective values outside any prescribed neighborhood, without assuming smoothness or strong convexity. Our analysis identifies a displacement condition as the key driver of linear descent and provides two complementary parameter regimes to guarantee it: fixing the trust-region radius and choosing the regularization properly, or fixing the regularization and selecting radii via a uniform displacement lower bound. We further give explicit characterization of the linear regime conditions respectively, and prove that the trust-region is redundant under strong convexity, Finally, we establish an exact equivalence with the Broximal Point Method (BPM) (Gruntkowska et al., 2025) in the active constraint regime.

Stabilized Proximal Point Method via Trust Region Control

Abstract

The Proximal Point Method (PPM) (Rockafellar, 1976) is a fundamental tool for nonsmooth convex optimization. However, its convergence is not linear under general convexity in the absence of strong convexity or other structural assumptions. To address this limitation, we study a trust-region stabilized proximal point scheme in which each proximal update is computed over a localized feasible region. We show that this simple stabilization enforces non-vanishing steps and yields a linear decrease in objective values outside any prescribed neighborhood, without assuming smoothness or strong convexity. Our analysis identifies a displacement condition as the key driver of linear descent and provides two complementary parameter regimes to guarantee it: fixing the trust-region radius and choosing the regularization properly, or fixing the regularization and selecting radii via a uniform displacement lower bound. We further give explicit characterization of the linear regime conditions respectively, and prove that the trust-region is redundant under strong convexity, Finally, we establish an exact equivalence with the Broximal Point Method (BPM) (Gruntkowska et al., 2025) in the active constraint regime.

Paper Structure

This paper contains 50 sections, 21 theorems, 148 equations, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. The many facets of PPM
  4. Optimizer Design.
  5. Distributed Optimization.
  6. Projections.
  7. Splitting Algorithms.
  8. Acceleration.
  9. Smoothing.
  10. BPM and Trust Region Method.
  11. Convergence Theory of TRPPM
  12. Fixing $t_k \equiv t$
  13. Fixing $\lambda_k \equiv \lambda$
  14. The quantity $m_f(\varepsilon, \lambda)$
  15. Strongly Convex Case
  16. ...and 35 more sections

Key Result

Lemma 4.3

Let $f : \mathbb{R}^d \to \mathbb{R} \cup \left\{ +\infty \right\}$ satisfy assmp:pcc, and consider the update equation eq:trppm-def with a fixed radius $t_k \equiv t > 0$. For any iterate $x_k$ such that $\operatorname{dist}(x_k,{\mathcal{X}}_\star^f) > t$, there exists $\lambda_k^\star > 0$ such t Moreover, for any $\lambda_k \leq \lambda_k^\star$, it holds that $\blacktriangleleft$$\blacktrian

Theorems & Definitions (59)

  • Definition 1.1
  • Example 1.2
  • Definition 3.1
  • Definition 4.2
  • Lemma 4.3: Existence of $\lambda$
  • Theorem 4.4
  • Lemma 4.5
  • Lemma 4.7
  • Corollary 4.8
  • Remark 4.9: Comparison with PPM
  • ...and 49 more