An O(m/ε^3.5)-Cost Algorithm for Semidefinite Programs with Diagonal Constraints
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We study semidefinite programs with diagonal constraints. This problem class appears in combinatorial optimization and has a wide range of engineering applications such as in circuit design, channel assignment in wireless networks, phase recovery, covariance matrix estimation, and low-order controller design. In this paper, we give an algorithm to solve this problem to ε-accuracy, with a run time of O(m/ε^3.5), where m is the number of non-zero entries in the cost matrix. We improve upon the previous best run time of O(m/ε^4.5) by Arora and Kale. As a corollary of our result, given an instance of the Max-Cut problem with n vertices and m ≫ n edges, our algorithm when applied to the standard SDP relaxation of Max-Cut returns a (1 - ε)α_GW cut in time O(m/ε^3.5), where α_GW≈ 0.878567 is the Goemans-Williamson approximation ratio. We obtain this run time via the stochastic variance reduction framework applied to the Arora-Kale algorithm, by constructing a constant-accuracy estimator to maintain the primal iterates.
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