An \(\widetilde{\mathcal{O}}(m/\varepsilon^{3.5})\)-Cost Algorithm for Semidefinite Programs with Diagonal Constraints

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 \(\varepsilon\)-accuracy, with a run time of \(\widetilde{\mathcal{O}}(m/\varepsilon^{3.5})\), where \(m\) is the number of non-zero entries in the cost matrix. We improve upon the previous best run time of \(\widetilde{\mathcal{O}}(m/\varepsilon^{4.5})\) by Arora and Kale. As a corollary of our result, given an instance of the Max-Cut problem with \(n\) vertices and \(m \gg n\) edges, our algorithm when applied to the standard SDP relaxation of Max-Cut returns a \((1 - \varepsilon)\alpha_{GW}\) cut in time \(\widetilde{\mathcal{O}}(m/\varepsilon^{3.5})\), where \(\alpha_{GW} \approx 0.878567\) 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.