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^\omega + n^\omega) \log(1 / \epsilon) ), \end{align*} 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 of Jiang, Lee, Song, and Wong [JLSW20]. Our algorithm's runtime can be naturally interpreted as follows: $\widetilde{O}(\sqrt{n} \log (1/\epsilon))$ 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.

中文翻译：

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半定规划的一种更快的内点法

半定规划 (SDP) 是一类基本的优化问题，最近在近似算法、量子复杂性、鲁棒学习、算法舍入和对抗性深度学习中具有重要的应用。本文提出了一种更快的内点方法来解决具有可变大小 $n \times n$ 和 $m$ 时间约束的泛型 SDPs \begin{align*} \widetilde{O}(\sqrt{n}( mn^2 + m^\omega + n^\omega) \log(1 / \epsilon) ), \end{align*} 其中 $\omega$ 是矩阵乘法的指数，$\epsilon$ 是相对精度。在 $m \geq n$ 的主要情况下，我们的运行时间优于之前最快的 SDP 求解器，后者基于 Jiang、Lee、Song 和 Wong 的切割平面方法 [JLSW20]。我们算法的运行时间可以很自然地解释如下：$\widetilde{O}(\sqrt{n} \log (1/\epsilon))$ 是我们的内点法所需的迭代次数，$mn^2$ 是输入大小，$m^\omega + n^\omega$ 是在每次迭代中反转 Hessian 和 slack 矩阵的时间。这些构成了进一步改进用于解决通用 SDP 的内点方法的运行时间的天然障碍。