Stabilized Proximal Point Method via Trust Region Control

TL;DR

This paper introduces a trust-region stabilized proximal point method that ensures linear convergence in nonsmooth convex optimization without requiring strong convexity or smoothness.

Contribution

It proposes a simple stabilization technique using a trust-region approach that guarantees linear decrease in objective values under broad conditions.

Findings

Enforces non-vanishing steps for linear convergence

Provides explicit parameter regimes for stability

Establishes equivalence with Broximal Point Method in certain regimes

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.