This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. while the latter was initially proposed by Wang et al. and later exploited in the TSSOS hierarchy. In doing so we obtain CS-TSSOS -- a two-level hierarchy of semidefinite programming relaxations with (i), the crucial property to involve quasi block-diagonal matrices and (ii), the guarantee of convergence to the global optimum. We demonstrate its efficiency on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to several thousands of variables and ten thousands of constraints.

中文翻译：

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CS-TSSOS：大规模多项式优化的相关性和项稀疏性

这项工作提出了一种新的矩 SOS 层次结构，称为 CS-TSSOS，用于解决大规模稀疏多项式优化问题。它的新颖之处在于通过结合两个现有稀疏多项式优化框架的优点，同时利用相关稀疏性和项稀疏性。前者是由于 Waki 等人。而后者最初是由Wang等人提出的。后来在 TSSOS 层次结构中被利用。通过这样做，我们获得了 CS-TSSOS——半定规划松弛的两级层次结构，具有（i），涉及准块对角矩阵的关键特性和（ii），收敛到全局最优的保证。我们在著名的 Max-Cut 问题和重要的工业优化潮流问题的几个大规模实例上证明了它的效率，