A Faster Interior Point Method for Semidefinite Programming

Abstract

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 $n \times n$ and m constraints in time \begin{equation*} \tilde{O}(\sqrt{n}(mn^{2}+m^{\omega}+n^{\omega})\log(1/\epsilon)), \end{equation*} where $\omega$ is the exponent of matrix multiplication and $\epsilon$ is the relative accuracy. In the predominant case of $m\geq n$ , 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: $O(\sqrt{n}\log(1/\epsilon))$ is the number of iterations needed for our interior point method, $mn^{2}$ is the input size, and $m^{\omega}+n^{\omega}$ 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.