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We study the design of polylogarithmic depth algorithms for approximatelysolving packing and covering semidefinite programs (or positive SDPs forshort). This is a natural SDP generalization of the well-studied positive LPproblem.  Although positive LPs can be solved in polylogarithmic depth while using only$\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best knownpositive SDP solvers due to Jain and Yao require $O(\log^{14} n/\varepsilon^{13})$ parallelizable iterations. Several alternative solvers havebeen proposed to reduce the exponents in the number of iterations. However, thecorrectness of the convergence analyses in these works has been called intoquestion, as they both rely on algebraic monotonicity properties that do notgeneralize to matrix algebra.  In this paper, we propose a very simple algorithm based on the optimizationframework proposed for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n/ \varepsilon^2)$ iterations, matching that of the best LP solver. To surmountthe obstacles encountered by previous approaches, our analysis requires a newmatrix inequality that extends Lieb-Thirring's inequality, and asign-consistent, randomized variant of the gradient truncation techniqueproposed in.

Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver