SIAM Journal on Optimization, Volume 31, Issue 2, Page 1433-1458, January 2021.
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. This algorithm does not require the solution to be strictly feasible and works for large problems. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the ${E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb{R}^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb{Q}[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.

中文翻译：

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包装问题的精确半定规划界限

SIAM Journal on Optimization，第 31 卷，第 2 期，第 1433-1458 页，2021 年 1 月。
在本文中，我们给出了一种算法，将半定规划求解器的浮点输出四舍五入为有理数的解或有理数的二次扩展。该算法不需要解决方案严格可行并且适用于大问题。我们应用它来获得包装问题的明确界限，并且我们使用这些明确界限来证明某些最佳包装配置在旋转之前是唯一的。特别地，我们表明来自 ${E}_8$ 根格子的配置是在 $\mathbb{R}^8$ 半球上具有最小角距离 $\pi/3$ 的唯一最优代码，我们证明$(3, 8, \vartheta)$-球面代码的三点界限，其中$\vartheta$满足$\cos \vartheta = (2\sqrt{2}-1)/7$ , 通过四舍五入到 $\mathbb{Q}[\sqrt{2}]$ 变得尖锐。