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
$\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known
positive SDP solvers due to Jain and Yao require $O(\log^{14} n
/\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have
been proposed to reduce the exponents in the number of iterations. However, the
correctness of the convergence analyses in these works has been called into
question, 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 for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n
/ \varepsilon^2)$ 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.

本文针对包装和覆盖半正定程序（或正半定规划）的近似求解，研究了多对数深度算法的设计，并提出了一种基于优化框架的简单算法，其正确性基于 Lieb-Thirring 不等式的新矩阵不等式和梯度截断技术的随机变体。