An Overview and Comparison of Spectral Bundle Methods for Primal and Dual Semidefinite Programs

TL;DR

This paper provides an overview and comparison of spectral bundle methods for primal and dual SDPs, introducing a new primal-focused method that achieves linear convergence and demonstrates superior scalability in large-scale polynomial optimization.

Contribution

The paper introduces a new spectral bundle method for primal SDPs, complementing existing dual methods, with proven linear convergence and improved scalability for large problems.

Findings

The new primal spectral bundle method achieves linear convergence for primal feasibility, dual feasibility, and duality gap.

The methods are effective for SDPs with low-rank primal or dual solutions, respectively.

Numerical experiments show state-of-the-art efficiency in large-scale polynomial optimization.

Abstract

The spectral bundle method developed by Helmberg and Rendl is well-established for solving large-scale semidefinite programs (SDPs) in the dual form, especially when the SDPs admit $\textit{low-rank primal solutions}$. Under mild regularity conditions, a recent result by Ding and Grimmer has established fast linear convergence rates when the bundle method captures $\textit{the rank of primal solutions}$. In this paper, we present an overview and comparison of spectral bundle methods for solving both $\textit{primal}$ and $\textit{dual}$ SDPs. In particular, we introduce a new family of spectral bundle methods for solving SDPs in the $\textit{primal}$ form. The algorithm developments are parallel to those by Helmberg and Rendl, mirroring the elegant duality between primal and dual SDPs. The new family of spectral bundle methods also achieves linear convergence rates for primal feasibility, dual feasibility, and duality gap when the algorithm captures $\textit{the rank of the dual solutions}$. Therefore, the original spectral bundle method by Helmberg and Rendl is well-suited for SDPs with $\textit{low-rank primal solutions}$, while on the other hand, our new spectral bundle method works well for SDPs with $\textit{low-rank dual solutions}$. These theoretical findings are supported by a range of large-scale numerical experiments. Finally, we demonstrate that our new spectral bundle method achieves state-of-the-art efficiency and scalability for solving polynomial optimization compared to a set of baseline solvers $\textsf{SDPT3}$, $\textsf{MOSEK}$, $\textsf{CDCS}$, and $\textsf{SDPNAL+}$.