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

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 Lov\'{a}sz 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 $10^{-2}$.