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

TL;DR

The paper addresses scalable semidefinite programming by developing a primal spectral bundle method that mirrors well-known dual approaches. It formulates primal SDPs as exact-penalty eigenvalue problems and introduces the $(r_p,r_c)$-SBMP family, which uses a low-rank spectral basis to build improved lower models and small master problems. Theoretical results establish sublinear convergence $O(1/ ext{gap}^3)$ generally, with linear convergence under strict complementarity and rank conditions, and a detailed comparison to dual spectral bundle methods clarifies when each approach excels. Numerically, the method demonstrates state-of-the-art efficiency and scalability for polynomial optimization relaxations (SOS, Max-Cut) compared to SDPT3, MOSEK, CDCS, and SDPNAL+. Overall, the work broadens spectral bundle optimization to primal SDPs and provides practical open-source implementations for large-scale SDP challenges.

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 $\textsf{SDPNAL+}$.

Figures (2)

• Figure 1: The relative optimality gap of different choices of $r_{\mathrm{p}}$ and $r_{\mathrm{c}}$ in $(r_\mathrm{p},r_\mathrm{c})$-SBMP and $(r_\mathrm{p},r_\mathrm{c})$-SBMD for solving two SDPs with $X \in \mathbb{S}^{1000}_+$: the left SDP instance admits a low-rank dual solution $\mathrm{rank}(Z^\star) = 3$ which suits well for $(r_\mathrm{p},r_\mathrm{c})$-SBMP, while the right SDP instance admits a low-rank primal solution $\mathrm{rank}(X^\star) = 3$ that suits well for $(r_\mathrm{p},r_\mathrm{c})$-SBMD.
• Figure 2: The relative optimality gap of different choices of $r_{\mathrm{p}}$ and $r_{\mathrm{c}}$ in $(r_\mathrm{p},r_\mathrm{c})$-SBMP and $(r_\mathrm{p},r_\mathrm{c})$-SBMD for solving SDP relaxations of two Max-Cut problems.

Theorems & Definitions (52)

• Proposition 2.1: ding2020revisit
• Lemma 2.1: alizadeh1997complementarity
• Definition 2.1: alizadeh1997complementarity
• Theorem 2.1: ruszczynski2011nonlinear
• Lemma 2.2: diaz2023optimal
• Proposition 3.1
• Proposition 3.2
• Remark 3.1: Exact penalization for primal SDPs
• Proposition 3.3
• proof