Historically, scalability has been a major challenge for the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this article, we survey recent approaches to this challenge, including those that exploit structure (e.g., sparsity and symmetry) in a problem, those that produce low-rank approximate solutions to semidefinite programs, those that use more scalable algorithms that rely on augmented Lagrangian techniques and the alternating-direction method of multipliers, and those that trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs). For each class of approaches, we provide a high-level exposition, an entry point to the corresponding literature, and examples drawn from machine learning, control, or robotics. We also present a list of software packages that implement many of the techniques discussed in the review. Our hope is that this article will serve as a gateway to the rich and exciting literature on scalable semidefinite programming for both theorists and practitioners.
Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope