A unified analysis of a class of proximal bundle methods for solving hybrid convex composite optimization problems

TL;DR

This paper introduces a unified proximal bundle framework for hybrid convex composite optimization, providing iteration-complexity bounds for multiple variants and proposing a universal adaptive method with comparable complexity.

Contribution

It establishes a common complexity analysis for various proximal bundle methods and introduces a new universal adaptive variant based on one-cut models.

Findings

Unified iteration-complexity bounds for three PB variants.

First unified complexity analysis for these PB variants in HCCO.

Proposed universal adaptive PB has the same complexity as other variants.

Abstract

This paper presents a proximal bundle (PB) framework based on a generic bundle update scheme for solving the hybrid convex composite optimization (HCCO) problem and establishes a common iteration-complexity bound for any variant belonging to it. As a consequence, iteration-complexity bounds for three PB variants based on different bundle update schemes are obtained in the HCCO context for the first time and in a unified manner. While two of the PB variants are universal (i.e., their implementations do not require parameters associated with the HCCO instance), the other newly (as far as the authors are aware of) proposed one is not but has the advantage that it generates simple, namely one-cut, bundle models. The paper also presents a universal adaptive PB variant (which is not necessarily an instance of the framework) based on one-cut models and shows that its iteration-complexity is the same as the two aforementioned universal PB variants.