SDP relaxation method for detecting P-tensors

Abstract

P-tensor and P 0 -tensor are introduced in tensor complementarity problem, which have wide applications in many fields such as game theory, tensor complementarity problem. In this paper, we discuss how to check whether a given symmetric tensor is P(P 0 )-tensor or not. For a symmetric tensor, it is a P(P 0 )-tensor is equivalent to the positivity(nonnegativity) of a polynomial optimization problem. For such polynomial optimization problem, a SDP relaxation method is proposed. By the proposed method, the P(P 0 )-tensor can be detected by solving a finite number of SDP relaxations. Furthermore, numerical examples are reported to show the efficiency of the proposed algorithm.