SDP relaxation method for detecting P-tensors

Li Li,Xiao Wang

doi:10.19139/soic.v5i4.324

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.

Highlights

For positive integers m and n1, n2, · · ·, nm, an m-order and (n1, n2, · · ·, nm)-dimensional real tensor is an array in the space Rn1×n2×···×nm
The tensor space Rn1×n2×···×nm is denoted as Tm(Rn)
In this paper, we propose a numerical method by solving SDP relaxations to check whether the given symmetric tensor is P(P0)-tensor or not

Summary

Introduction

For positive integers m and n1, n2, · · · , nm, an m-order and (n1, n2, · · · , nm)-dimensional real tensor is an array in the space Rn1×n2×···×nm. SDP relaxation method is applied to compute Z(H)-eigenvalues in [1] Motivated by these fact, in this paper, we propose a numerical method by solving SDP relaxations to check whether the given symmetric tensor is P(P0)-tensor or not. In this paper, we propose a numerical method by solving SDP relaxations to check whether the given symmetric tensor is P(P0)-tensor or not It is a P-tensor if the minimum of SDP relaxation is positive; it is a P0-tensor but not P-tensor if the optimal value of the polynomial optimization problem is zero; it is not a P0-tensor if the optimal value of the polynomial optimization problem is negative.

Preliminaries

Numerical Examples

Conclusion