Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider t...

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Bibliographic Details
Published inScience China. Mathematics Vol. 58; no. 1; pp. 197 - 212
Main Authors Luo, ZiYan, Qi, LiQun, Ye, YinYu
Format Journal Article
LanguageEnglish
Published Heidelberg Science China Press 01.01.2015
Subjects
Applications of Mathematics
Frobenius范数
Mathematics
Mathematics and Statistics
对称半正定
对称张量
张量空间
性能表征
物理科学
线性算子
视锥细胞
symmetric tensor
linear operator
53C35
47A05
symmetric positive semidefinite tensor cone
53A45
15A69
SOS cone
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ISSN1674-7283
1869-1862
DOI10.1007/s11425-014-4930-z

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Summary:We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor. Furthermore, we characterize the symmetric positive semidefinite tensor （SDT） cone by employing the properties of linear operators, design some face structures of its dual cone, and analyze its relationship to many other tensor cones. In particular, we show that the cone is self-dual if and only if the polynomial is quadratic, give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases, and develop a complete relationship map among the tensor cones appeared in the literature.
Bibliography:symmetric tensor, symmetric positive semidefinite tensor cone, linear operator, SOS cone
We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor. Furthermore, we characterize the symmetric positive semidefinite tensor （SDT） cone by employing the properties of linear operators, design some face structures of its dual cone, and analyze its relationship to many other tensor cones. In particular, we show that the cone is self-dual if and only if the polynomial is quadratic, give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases, and develop a complete relationship map among the tensor cones appeared in the literature.
11-1787/N
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-014-4930-z