  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 $\textsf{SDPNAL+}$.
