Solving Tall Dense SDPs in the Current Matrix Multiplication Time

This paper introduces a new interior point method algorithm that solves semidefinite programming (SDP) with variable size $n \times n$ and $m$ constraints in the (current) matrix multiplication time $m^{\omega}$ when $m \geq \Omega(n^2)$. Our algorithm is optimal because even finding a feasible matrix that satisfies all the constraints requires solving an linear system in $m^{\omega}$ time. Our work improves the state-of-the-art SDP solver [Jiang, Kathuria, Lee, Padmanabhan and Song, FOCS 2020], and it is the first result that SDP can be solved in the optimal running time. Our algorithm is based on two novel techniques: $\bullet$ Maintaining the inverse of a Kronecker product using lazy updates. $\bullet$ A general amortization scheme for positive semidefinite matrices.