Linear operators and positive semidefiniteness of symmetric tensor spaces

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• 作者：ZiYan Luo (1)
  LiQun Qi (2)
  YinYu Ye (3)
  
  1. State Key Laboratory of Rail Traffic Control and Safety ; Beijing Jiaotong University ; Beijing ; 100044 ; China
  2. Department of Applied Mathematics ; The Hong Kong Polytechnic University ; Hung Hom ; Kowloon ; Hong Kong ; China
  3. Department of Management Science and Engineering ; Stanford University ; Stanford ; CA ; 94305 ; USA
  
• 关键词：symmetric tensor ; symmetric positive semidefinite tensor cone ; linear operator ; SOS cone ; 15A69 ; 53A45 ; 47A05 ; 53C35
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：58
• 期：1
• 页码：197-212
• 全文大小：283 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

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.