We study the design of polylogarithmic depth algorithms for approximately solving packing and covering semidefinite programs (or positive SDPs for short). This is a natural SDP generalization of the well-studied positive LP problem. Although positive LPs can be solved in polylogarithmic depth while using only log 2 n/ε 3 parallelizable iterations [3], the best known positive SDP solvers due to Jain and Yao [18] require log 14 n/ε 13 parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations [19, 29] . However, the correctness of the convergence analyses in these works has been called into question [29] , as they both rely on algebraic monotonicity properties that do not generalize to matrix algebra. In this paper, we propose a very simple algorithm based on the optimization framework proposed in [3] for LP solvers. Our algorithm only needs log 2 n/ε 3 iterations, matching that of the best LP solver. To surmount the obstacles encountered by previous approaches, our analysis requires a new matrix inequality that extends Lieb-Thirring's inequality, and a sign-consistent, randomized variant of the gradient truncation technique proposed in [2, 3] .
The state-of-the-art positive SDP solver was proposed by REF with only O(log 2 (mn)/ 3 ) iterations.
Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver
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