Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov–Poincaré operators.

中文翻译：

---

用于拓扑优化的箭头型正半定矩阵分解

Kojima 等人最近使用了具有弦稀疏图的矩阵的大矩阵不等式的分解。（Math Program 129(1):33–68, 2011）减少大规模半定优化 (SDO) 问题的问题规模，从而提高标准 SDO 软件的效率。这种分解的副产品是引入了新的密集小尺寸矩阵变量。我们将证明，对于满足适当假设的箭头类型矩阵，附加矩阵变量的秩为 1，因此可以用相同维度的向量变量替换。这会显着提高标准 SDO 软件的效率。我们将把这个思想应用到拓扑优化问题中，该问题被表述为一个大规模线性半定优化问题。与原始的大规模问题相比，数值示例将证明在分解问题的解决方案方面有巨大的加速。在我们的数值示例中，与原始问题公式中超过三次的增长相比，分解的问题表现出复杂性的线性增长。我们还将把我们的方法与域分解的标准理论联系起来，并表明额外的向量变量是相应的离散 Steklov-Poincaré 算子的结果。