A Faster Interior Point Method for Semidefinite Programming

Semidefinite programs (SDPs) are a fundamental class of optimization problems with important recent applications in approximation algorithms, quantum complexity, robust learning, algorithmic rounding, and adversarial deep learning. This paper presents a faster interior point method to solve generic SDPs with variable size n ×n and m constraints in time Õ(√n(mn ² +m ^ω +n ^ω )log(1/ε)), \end{equation*} where ω is the exponent of matrix multiplication and ε is the relative accuracy. In the predominant case of m ≥ n, our runtime outperforms that of the previous fastest SDP solver, which is based on the cutting plane method [JLSW20]. Our algorithm's runtime can be naturally interpreted as follows: O(√nlog(1/ε)) is the number of iterations needed for our interior point method, mn ² is the input size, and m ^ω +n ^ω is the time to invert the Hessian and slack matrix in each iteration. These constitute natural barriers to further improving the runtime of interior point methods for solving generic SDPs.