A New Eigenvalue For Tensor-Matrix And Related Complementarity Problem

This article is divided into two parts.In the first part,we propose a new defined eigenvalue(N-eigenvalue)problem for tensor and matrix,which arises from a very fa-mous problem in physics–the BEC problem.We study its general properties,including the boundary of eigenvalues,and solve it by semi-definite programming,under certain assumptions,we prove that relaxation is tight.In addition,we transform the problem into a polynomial optimization problem with constraint conditions and get all eigen-values with Lasserre's hierarchy of semidefinite relaxations.In the second part,gen-eralized eigenvalue complementarity problem for tensors and matrix(GEiCP-T M)_J is studied.We established the relationship between it and the above eigenvalue prob-lem and prove that the number of solutions is bounded.Some sufficient conditions for its existence are given.For symmetric case,we establish a relationship with the opti-mization problem and derive the necessary and sufficient conditions for its solution.In addition,we apply the adaptive projection method and the above Lasserre's hierarchy of semidefinite relaxations to solve this problem respectively,and get a pair of solution and all pairs of solution for(EiCP-T M).The convergence of the above algorithms is analyzed and some numerical examples are given.