David Fersztand | Searcharxiv

arXiv2026

An Absolute-Error Proximal Bundle Method through the Lens of Frank-Wolf

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving optimization problems with nonsmooth components. In this paper, we conduct a theoretical investigation of a modified proximal bundle method, which we call the Modified Proximal Bundle with Fixed Absolute Accuracy (MPB-FA). MPB-FA modifies the serious-step test of the PBM serious-step test. In MPB-FA, it is based on an absolute accuracy criterion, and the accuracy is fixed over iterations, while the standard PBM uses a relative accuracy in the serious-step test, which changes with iterations. Also, similarly to multiple PBM analyses, the proximal parameter in MPB-FA is also fixed over iterations, while it is permitted to change in the standard PBM. These modifications allow us to build the first link between a proximal bundle method and a Frank-Wolfe algorithm on the Moreau envelope of the dual problem. In light of this correspondence, we first extend the linear convergence of Kelley's method on the sum of a smooth strongly convex function and a convex piecewise linear function from the positive homogeneous to the general case. Building on this result, we propose a novel complexity analysis of MPB-FA when the objective is the sum of a smooth and a piecewise function and derive an $\mathcal{O}(ε^{-8/9})$ iteration complexity, improving upon the best known $\mathcal{O}(ε^{-2})$ guarantee on a related variant of PBM. It is worth-noting that the best known complexity bound for the classical PBM in the general case is $\mathcal{O}(ε^{-3})$ and $\mathcal{O}(ε^{-2})$ when the proximal parameter is fixed. Our approach also reveals new insights on bundle management and empirical behavior of the proximal bundle methods.

math.OC↗

arXiv2025

On the Acceleration of Proximal Bundle Methods

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving nonsmooth optimization problems. In this paper, we present the first variant of the PBM for smooth objectives, achieving an accelerated convergence rate of $O(ε^{-1/2}\log(1/ε))$, where $ε$ is the desired accuracy. Our approach addresses an open question regarding the convergence guarantee of proximal bundle type methods, which was previously posed in two recent papers. We interpret the PBM as a proximal point algorithm and base our proposed algorithm on an accelerated inexact proximal point scheme. Our variant introduces a novel null step test and oracle while maintaining the core structure of the original algorithm. The newly proposed oracle substitutes the traditional cutting planes with a smooth lower approximation of the true function. We show that this smooth interpolating lower model can be computed as a convex quadratic program. We also examine a second setting where Nesterov acceleration can be effectively applied, specifically when the objective is the sum of a smooth function and a piecewise linear one.

math.OC↗