A Faster Interior Point Method for Semidefinite Programming
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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 × n and m constraints in time O(√(n)( mn^2 + m^ω + n^ω) log(1 / ϵ) ), where ω is the exponent of matrix multiplication and ϵ is the relative accuracy. In the predominant case of m ≥ 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: O(√(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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