Test Of Two Classes Of Structure Tensors

With the development of tensor analysis and tensor computation,the structure tensor is becoming more and more important in tensor research.One class of structure tensor is copositive tensor,which has widely applications in polynomial optimization,hypergraph,tensor eigenvalue complementarity,etc.Another class of structure tensor is symmetric P tensor,which has important applications in the solution structure of tensor complementary and tensor variational inequality.Therefore,it is important to determine whether a given tensor is copositive tensor（symmetric P tensor）or not.In this paper,we establish SDP relaxation algorithms to test copositivity and P-property.Firstly,two classes of problems are equivalently reformulated as the corresponding polynomial op-timization problems,respectively.Next,the corresponding positive semidefinite（SDP）relaxation algorithms are respectively proposed based on reformulated polynomial op-timization problems.Finally,by solving SDP relaxation sequentially,the optimal val-ues of the reformulated polynomial optimization problems are obtained,and hence the structure tensor are tested respectively.Specifically,the tested tensor is strictly coposi-tive（symmetric P）tensor if the optimal value of one of SDP relaxation is positive;the tested tensor is copositive but not strictly copositive（symmetric P₀but not P）tensor if the optimal value is zero;the tested tensor is not copositive（symmertic P₀）tensor if the optimal value is negative.In this paper,the corresponding algorithms of two classes of structure tensors have finite convergence under certain conditions.Numerical result-s respectively show the effectiveness of the proposed corresponding algorithm in the paper.