实对称矩阵$m$的同余同时对角化及其在optimization中的意义
Simultaneous diagonalization via congruence of $m$ real symmetric matrices and its implications in optimization
摘要：
令$\{C_1, C_2, \ldots, C_m\},~m\ge2$是$n\times n$实对称矩阵的集合。本文的目的是提供一个算法，找到一个共同的同余矩阵$R$，使$R^TC_iR$是真正的对角线为每个$C_i;$或报告没有这样的。这个问题被称为通过同余同时对角化（简称SDC），乍一看似乎是纯线性代数的问题。然而，对于二次约束二次规划（QCQP），如果二次型是SDC，它们的联合范围集是一个封闭凸多面锥，这打开了可能性，以扩展经典的$\mathcal{S}$ -引理的两个以上的对称矩阵。此外，在二次形式的SDC假设下，QCQP可以被改写为通常更容易处理的可分离形式。因此，重要的是要有一个标准的程序，以确定是否SDC属性为下划线的二次优化问题。我们的结果解决了Hiriart-Urruty在2007年提出的一个长期存在的问题。
英文摘要：
Let $\{C_1, C_2, \ldots, C_m\},~m\ge2$ be a collection of $n\times n$ real symmetric matrices. The objective of the paper is to offer an algorithm that finds a common congruence matrix $R$ such that $R^TC_iR$ is real diagonal for every $C_i;$ or reports none of such kind. The problem, referred to as the simultaneously diagonalization via congruence (SDC in short), seems to be of pure linear algebra at first glance. However, for quadratically constrained quadratic programming (QCQP), if the quadratic forms are SDC, their joint range set is a closed convex polyhedral cone, which opens the possibility to extend the classical $\mathcal{S}$-lemma for more than two symmetric matrices. In addition, under the SDC assumption of quadratic forms, QCQP can be recast in separable forms which is usually easier to tackle. It is thus important to have a standard procedure for determining whether or not the SDC property holds for the underlined quadratic optimization problem. Our result solves a long standing problem posed by Hiriart-Urruty in 2007.
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