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{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.