An Accelerated Proximal Bundle Method with Momentum

TL;DR

This paper introduces an accelerated proximal bundle method that incorporates Nesterov's momentum, achieving the optimal convergence rate for smooth convex optimization problems.

Contribution

The paper proposes a novel accelerated proximal bundle method that attains the optimal O(1/k^2) convergence rate by integrating Nesterov's momentum into traditional PBM.

Findings

APBM achieves the optimal O(1/k^2) convergence rate.

Numerical experiments confirm the effectiveness of APBM.

APBM outperforms standard PBM in smooth convex problems.

Abstract

Proximal bundle methods (PBM) are a powerful class of algorithms for convex optimization. Compared to gradient descent, PBM constructs more accurate surrogate models that incorporate gradients and function values from multiple past iterations, which leads to faster and more robust convergence. However, for smooth convex problems, PBM only achieves an O(1/k) convergence rate, which is inferior to the optimal O(1/k^2) rate. To bridge this gap, we propose an accelerated proximal bundle method (APBM) that integrates Nesterov's momentum into PBM. We prove that under standard assumptions, APBM achieves the optimal O(1/k^2) convergence rate. Numerical experiments demonstrate the effectiveness of the proposed APBM.