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An accelerated proximal bundle method for convex optimization

Feng-Yi Liao, Thomas Madden, Yang Zheng

TL;DR

The paper tackles accelerating the proximal bundle method (PBM) for convex optimization by integrating Nesterov-style momentum into the PBM framework. It treats the PBM inner proximal step as an inexact (sub)gradient update and shows that, with appropriate parameters, the method attains the optimal $\mathscr{O}(1/\sqrt{\epsilon})$ rate for smooth convex objectives, while preserving the classical bundle-model assumptions. The authors establish a theoretical link to accelerated inexact proximal point methods and provide a rigorous $\mathscr{O}(1/\sqrt{\epsilon})$ complexity bound, along with $O(1/k^2)$ convergence in the smooth regime. Numerical experiments corroborate the theory, demonstrating that APBM outperforms classical PBM and Nesterov's AGD across tested problems and parameter settings. Overall, this work offers a practical, principled path to speed PBM in smooth settings and invites further extensions to broader nonsmooth contexts.

Abstract

The proximal bundle method (PBM) is a powerful and widely used approach for minimizing nonsmooth convex functions. However, for smooth objectives, its best-known convergence rate remains suboptimal, and whether PBM can be accelerated remains open. In this work, we present the first accelerated proximal bundle method that achieves the optimal $\mathscr{O}(1/\sqrtε)$ iteration complexity for obtaining an $ε$-accurate solution in smooth convex optimization. The proposed method is conceptually simple, which differs from Nesterov's accelerated gradient descent by only a single line and retains all key structural properties of the classical PBM. In particular, it relies on the same minimal assumptions on model approximations and preserves the standard bundle testing criterion. Numerical experiments confirm the accelerated $\mathscr{O}(1/\sqrtε)$ convergence rate predicted by our theory.

An accelerated proximal bundle method for convex optimization

TL;DR

The paper tackles accelerating the proximal bundle method (PBM) for convex optimization by integrating Nesterov-style momentum into the PBM framework. It treats the PBM inner proximal step as an inexact (sub)gradient update and shows that, with appropriate parameters, the method attains the optimal rate for smooth convex objectives, while preserving the classical bundle-model assumptions. The authors establish a theoretical link to accelerated inexact proximal point methods and provide a rigorous complexity bound, along with convergence in the smooth regime. Numerical experiments corroborate the theory, demonstrating that APBM outperforms classical PBM and Nesterov's AGD across tested problems and parameter settings. Overall, this work offers a practical, principled path to speed PBM in smooth settings and invites further extensions to broader nonsmooth contexts.

Abstract

The proximal bundle method (PBM) is a powerful and widely used approach for minimizing nonsmooth convex functions. However, for smooth objectives, its best-known convergence rate remains suboptimal, and whether PBM can be accelerated remains open. In this work, we present the first accelerated proximal bundle method that achieves the optimal iteration complexity for obtaining an -accurate solution in smooth convex optimization. The proposed method is conceptually simple, which differs from Nesterov's accelerated gradient descent by only a single line and retains all key structural properties of the classical PBM. In particular, it relies on the same minimal assumptions on model approximations and preserves the standard bundle testing criterion. Numerical experiments confirm the accelerated convergence rate predicted by our theory.

Paper Structure

This paper contains 21 sections, 14 theorems, 66 equations, 4 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statement
  3. Gradient descent and proximal point methods
  4. Proximal bundle method as an inexact PPM
  5. Problem statement
  6. An accelerated proximal bundle method
  7. Integrating Nesterov's extrapolation with PBM
  8. Accelerated rate for smooth convex problems
  9. Connections with Nesterov's AGD
  10. Technical proofs
  11. A simplified A-HPE algorithm
  12. Proof of \ref{['thm:A-PBM']}
  13. Numerical experiments
  14. Conclusion
  15. Additional properties of the proximal bundle method
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose $f$ is convex and $M$-smooth. Let $\beta \in (0,1)$ and $\rho>0$. Then, for any $y_k \in \mathbb{R}^n$, ProxDescent($y_k,\beta,\rho$) in alg:Proxi-descent-subproblem terminates in at most $\frac{16(M+\rho)^3}{(1-\beta)^2 \rho^3}$ iterations. Accordingly, the total iteration complexity of alg

Figures (4)

  • Figure 1: Numerical experiments. The worst-case function is taken from nesterov2018lectures, and APBM denotes the accelerated PBM in \ref{['alg:A-PBM-main']}.
  • Figure : Proximal Bundle method (PBM)
  • Figure : $\mathtt{ProxDescent}$($y_k,\beta,\rho$)
  • Figure : Simplified A-HPE

Theorems & Definitions (21)

  • Lemma 1: diaz2023optimal
  • Theorem 1
  • Remark 1
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • proof
  • Corollary 1
  • proof
  • theorem 1: liao2025bundle
  • ...and 11 more