CS-TSSOS：大规模多项式optimization的相关性和项稀疏性
CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization
摘要：
本文提出了一种新的矩SOS层次结构，称为CS-TSSOS，用于求解大规模稀疏多项式最优化问题。它的新颖之处是同时利用相关稀疏性和术语稀疏性，结合现有的两个框架的优势稀疏多项式优化。前者是由Waki等人提出的，而后者最初是由Wang等人提出的，后来在TSSOS层次中得到了利用。在这样做，我们得到CS-TSSOS -一个两级层次的半定规划松弛与（i），涉及块的SDP矩阵的关键属性和（ii），保证在一定条件下收敛到全局最优值。我们证明了它的效率和可扩展性的几个大规模的情况下，著名的最大割问题和重要的工业最优潮流问题，涉及多达六千个变量和数万个约束。
英文摘要：
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 blocks of SDP matrices and (ii), the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints.
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