Solving Tall Dense SDPs in the Current Matrix Multiplication Time
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This paper introduces a new interior point method algorithm that solves semidefinite programming (SDP) with variable size n × n and m constraints in the (current) matrix multiplication time m^ω when m ≥Ω(n^2). Our algorithm is optimal because even finding a feasible matrix that satisfies all the constraints requires solving an linear system in m^ω 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: ∙ Maintaining the inverse of a Kronecker product using lazy updates. ∙ A general amortization scheme for positive semidefinite matrices.
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