Suboptimality bounds for trace-bounded SDPs enable a faster and scalable low-rank SDP solver SDPLR+

TL;DR

The paper addresses the scalability gap in semidefinite programming by introducing SDPLR+, a low-rank SDP solver that leverages a suboptimality bound derived from trace-bounded duality. By combining an augmented Lagrangian framework with dynamic rank updates and efficient eigenvalue computations, SDPLR+ achieves faster convergence and reduced memory usage on very large SDPs. Empirical results across Max Cut, Minimum Bisection, Lovász Theta, and Cut Norm problems demonstrate that SDPLR+ is often the fastest method to moderate accuracy and scales to problems with up to $10^6$ decision variables. This work offers a practical, scalable approach with broad applicability to graph-based SDP relaxations and related trace-bounded formulations.

Abstract

Semidefinite programs (SDPs) and their solvers are powerful tools with many applications in machine learning and data science. Designing scalable SDP solvers is challenging because by standard the positive semidefinite decision variable is an $n \times n$ dense matrix, even though the input is often $n \times n$ sparse matrices. However, the information in the solution may not correspond to a full-rank dense matrix as shown by Barvinok and Pataki. Two decades ago, Burer and Monteiro developed an SDP solver $\texttt{SDPLR}$ that optimizes over a low-rank factorization instead of the full matrix. This greatly decreases the storage cost and works well for many problems. The original solver $\texttt{SDPLR}$ tracks only the primal infeasibility of the solution, limiting the technique's flexibility to produce moderate accuracy solutions. We use a suboptimality bound for trace-bounded SDP problems that enables us to track the progress better and perform early termination. We then develop $\texttt{SDPLR+}$, which starts the optimization with an extremely low-rank factorization and dynamically updates the rank based on the primal infeasibility and suboptimality. This further speeds up the computation and saves the storage cost. Numerical experiments on Max Cut, Minimum Bisection, Cut Norm, and Lovász Theta problems with many recent memory-efficient scalable SDP solvers demonstrate its scalability up to problems with million-by-million decision variables and it is often the fastest solver to a moderate accuracy of $10^{-2}$.

Figures (6)

• Figure 1: Results on Max Cut. We set tolerance $\varepsilon = 10^{-2}$, and test each solver on all 202 graphs. We time out each solver after 8 hours. Figure \ref{['fig:max-cut-scalability']} and \ref{['fig:max-cut-running-time']} display how the running time of each solver scales with the problem size $n$ and the running time performance profile plot of all the solvers respectively. Note that SDPLR+ is the fastest solver for most of the problem instances and is within a small factor of the optimal solver for the other instances. While SCAMS also exhibits good scalability, from Figure \ref{['fig:max-cut-cuts']}, we see that it usually produces smaller cuts.
• Figure 2: Results on Minimum Bisection. We set tolerance $\varepsilon = 10^{-2}$, and test each solver on all 202 graphs. We time out each solver after 8 hours. Figure \ref{['fig:minimum-bisection-scalability']} and \ref{['fig:minimum-bisection-dt']} display how the running time of each solver scales with the problem size $n$ and the running time performance profile plot of all the solvers respectively. Note that SDPLR+ is overall the fastest solver. The constrained Riemannian optimization methods fail on over half of the problems. From Figure \ref{['fig:minimum-bisection-cut']}, we see SDPLR+ often provides a better rounded minimum bisection than SketchyCGAL.
• Figure 3: Results on Lovász Theta and Cut Norm.
• Figure 4: Statistics of $n$ and the number of non-zeros of 202 matrices we test on.
• Figure 5: Comparing SDPLR+ against SDPLR for solving Max Cut on Gset graphs.

Theorems & Definitions (3)

• Theorem 1
• proof
• Corollary 2