Semidefinite Relaxation Algorithms For The Detection Of P(P0)-tensor

As an important structural tensor,P(P₀)-tensor is the generalization of P(P₀)-matrix.Since the P(P₀)-matrix plays an important role in complementarity problems and variational inequalities,the research of the P(P₀)-tensor has become an important topic at present.As the extension of the linear complementarity problem,the tensor complementarity problem is a special nonlinear complementarity problem.With the in-depth study of the basic theory,algorithm and related applications of the problem,we found thatwhen the tensor is a P tensor,there is a non-empty compact solution set for the tensor complementarity problem.Therefore,it is of great significance to study the detection of P(P₀)-tensor.It is NP-hard to detect P(P₀)-tensor.To effectively detect the tensor,scholars have developed many algorithms,which use the properties of their Z(H)-eigenvalues of sym-metric P(P₀)-tensors.But for the asymmetric tensor,the theory between the eigenvalue and the detection is not clear.In this paper,we establish SDP relaxation algorithms for detecting P(P₀)-tensor.Specifically,We reformulate P(P₀)-tensor detection problem as polynomial optimization problems.Then we propose the SDP relaxation algorithms for solving the reformulated polynomial optimization problems.Meanwhile,it is proved that the proposed algorithms have finite convergence under certain conditions.Numeri-cal examples are reported to show the efficiency of the proposed algorithms.This paper lays a foundation for the further study of tensor complementarity problem and tensor variational inequality problem.