A Faster Interior Point Method for Semidefinite Programming

Haotian Jiang, Tarun Kathuria, Yin Tat Lee, Swati Padmanabhan, Zhao Song

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n \times n$</tex> and m constraints in time \begin{equation*} \tilde{O}(\sqrt{n}(mn^{2}+m^{\omega}+n^{\omega})\log(1/\epsilon)), \end{equation*} where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\omega$</tex> is the exponent of matrix multiplication and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon$</tex> is the relative accuracy. In the predominant case of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m\geq n$</tex> , 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: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{n}\log(1/\epsilon))$</tex> is the number of iterations needed for our interior point method, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$mn^{2}$</tex> is the input size, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m^{\omega}+n^{\omega}$</tex> 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.