论文标题
通过$ m $真实对称矩阵的一致性同时对角线化及其在优化中的影响
Simultaneous diagonalization via congruence of $m$ real symmetric matrices and its implications in optimization
论文作者
论文摘要
令$ \ {C_1，C_2，\ ldots，C_M \}，〜M \ GE2 $是$ N \ times n $真实对称矩阵的集合。该论文的目的是提供一种算法，该算法找到一个通用的一致性矩阵$ r $，以便每$ c_i; $; $; $; $; $; $ r^tc_ir $都是真正的对角线。乍一看，这个问题被称为同时通过一致性（简称SDC）的对角度化（SDC）似乎是纯线性代数。但是，对于二次约束二次编程（QCQP），如果二次形式是SDC，则其关节范围集是一个封闭的凸多面部性锥体，这将使经典$ \ Mathcal {s s} $ - 散失的可能性比两个对称的矩阵扩展。另外，在二次形式的SDC假设下，QCQP可以以可分离的形式重铸，通常更容易解决。因此，必须制定标准程序来确定SDC属性是否适用于下划线的二次优化问题。我们的结果解决了Hiriart-Arututy在2007年提出的长期存在的问题。
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.
