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 $n \times n$ and $m$ constraints in time \begin{align*} \widetilde{O}(\sqrt{n}( mn^2 + m^ω+ n^ω) \log(1 / ε) ), \end{align*} where $ω$ is the exponent of matrix multiplication and $ε$ 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 of Jiang, Lee, Song, and Wong [JLSW20]. Our algorithm's runtime can be naturally interpreted as follows: $\widetilde{O}(\sqrt{n} \log (1/ε))$ 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.

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