论文标题
关于$ 5 \ times 5 $共同矩阵的圆锥的准确性近似值的精确性
On the Exactness of Sum-of-Squares Approximations for the Cone of $5\times 5$ Copositive Matrices
论文作者
论文摘要
我们研究了圆锥内部近似值的层次结构$ \ MATHCAL {k}^{（r）} _ n $（\ Mathbb {n} $ in \ Mathbb {n} $）用于共孔$ \ text {cop} {cop} _n $ parrilo（parrilo parrilo（结构式semidefinity of semidefiantion and parrilo）interys and semidefinite and semige geomity insial geomelty insial geomety geomety geomety，加利福尼亚理工学院，2001年）。众所周知，$ \ text {Cop} _4 = \ Mathcal {K}^{（0）} _ 4 $，而圆锥体的结合$ \ Mathcal {k}^{（r）} _ n $ copt $ cop of $ $ n \ geq 6 $。在这里，我们调查了其余的情况$ n = 5 $，其中所有极端射线均以希尔德布兰德（Hildebrand）（5 $ \ times $ 5 $ 5共同源锥的极端射线。线性代数及其应用，437（7）：1538----1547，2012）。我们表明，在$ \ text {cop} _5 $的极端射线中，喇叭矩阵$ h $及其正对角尺度在极端的射线中起着出色的作用。我们表明，等效$ \ text {cop} _5 = \ bigcup_ {r \ geq 0} \ Mathcal {k}^{（r）} _ 5 $在且仅当$ h $的对角度缩放$ h $属于$ h $属于$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k}^（k}^（r）} _ 5 $ for时， \ mathbb {n} $。作为证明的主要成分，我们基于多项式正方形的总和，介绍了$ \ text {cop} _n $的新的lasserre型锥形内部近似值。我们显示了他们指向锥体$ \ mathcal {k}^{（r）} _ n $的链接，并且我们使用一种优化方法，允许利用有限收敛的结果，以显示lasserre层次结构以显示新锥体中的成员。
We investigate the hierarchy of conic inner approximations $\mathcal{K}^{(r)}_n$ ($r\in \mathbb{N}$) for the copositive cone $\text{COP}_n$, introduced by Parrilo (Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis, California Institute of Technology, 2001). It is known that $\text{COP}_4=\mathcal{K}^{(0)}_4$ and that, while the union of the cones $\mathcal{K}^{(r)}_n$ covers the interior of $\text{COP}_n$, it does not cover the full cone $\text{COP}_n$ if $n\geq 6$. Here we investigate the remaining case $n=5$, where all extreme rays have been fully characterized by Hildebrand (The extreme rays of the 5 $\times$ 5 copositive cone. Linear Algebra and its Applications, 437(7):1538--1547, 2012). We show that the Horn matrix $H$ and its positive diagonal scalings play an exceptional role among the extreme rays of $\text{COP}_5$. We show that equality $\text{COP}_5=\bigcup_{r\geq 0} \mathcal{K}^{(r)}_5$ holds if and only if any positive diagonal scaling of $H$ belongs to $\mathcal{K}^{(r)}_5$ for some $r\in \mathbb{N}$. As a main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for $\text{COP}_n$, based on sums of squares of polynomials. We show their links to the cones $\mathcal{K}^{(r)}_n$, and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones.
