We give the first approximation algorithm for mixed packing and covering semidefinite programs (SDPs) with polylogarithmic dependence on width. Mixed packing and covering SDPs constitute a fundamental algorithmic primitive with recent applications in combinatorial optimization, robust learning, and quantum complexity. The current approximate solvers for positive semidefinite programming can handle only pure packing instances, and technical hurdles prevent their generalization to a wider class of positive instances. For a given multiplicative accuracy of $\epsilon$, our algorithm takes $O(\log^3(nd\rho) \cdot \epsilon^{-3})$ parallelizable iterations, where $n$, $d$ are dimensions of the problem and $\rho$ is a width parameter of the instance, generalizing or improving all previous parallel algorithms in the positive linear and semidefinite programming literature. When specialized to pure packing SDPs, our algorithm's iteration complexity is $O(\log^2 (nd) \cdot \epsilon^{-2})$, a slight improvement and derandomization of the state-of-the-art (Allen-Zhu et. al. '16, Peng et. al. '16, Wang et. al. '15). For a wide variety of structured instances commonly found in applications, the iterations of our algorithm run in nearly-linear time. In doing so, we give matrix analytic techniques for overcoming obstacles that have stymied prior approaches to this open problem, as stated in past works (Peng et. al. '16, Mahoney et. al. '16). Crucial to our analysis are a simplification of existing algorithms for mixed positive linear programs, achieved by removing an asymmetry caused by modifying covering constraints, and a suite of matrix inequalities whose proofs are based on analyzing the Schur complements of matrices in a higher dimension. We hope that both our algorithm and techniques open the door to improved solvers for positive semidefinite programming, as well as its applications.

翻译：我们给出了混合包装和覆盖半成品程序的第一个近似算法( SDPs ) 。 混合包装和覆盖 SDPs 构成了一个基本的算法原始, 最近在组合优化、 强力学习和量子复杂性方面的应用。 目前正半成品编程的近似解算器只能处理纯包装案例, 技术障碍可以防止它们被概括到更广泛的正面实例。 对于一个特定的纯包装SDPs来说, 我们的算法变异复杂性是 $( log3 (nd\rho)\ cdot delientral 3} $( cdot demodentral)\ central develop diversations, $ $( $ $, $daldalarnal) 和 $ $ $( $ $ $ $ $), 用于当前变异性变异性变变法, 也就是我们目前变现的变法技术的变现和变变变变变变法 。