A Class Of Tensor Generalized Higher Degree Eigenvalue Complementarity Problems

Tensor eigenvalue complementarity problem is a generalization of matrix eigenvalue complementarity problem.Usually the problem is transformed into an equivalent nonlinear programming problem because it is hard to solve.In this article,we introduce a class of tensor generalized higer-degree eigenvalue complementarity problem.And a necessary and sufficient condition for the stationary point of the corresponding nonlinear programming problem being the solution of the tensor generalized higher-degree eigenvalue complementarity problem is proved.Finally,under some conditions,the tensor generalized higher-degree eigenvalue complementarity problem can be transformed into the tensor higher-degree eigenvalue complementarity problem.Matrix eigenvalue complementarity problem is a kind of special eigenvalue problem,and is also a special class of complementarity problem.The complementarity problem is closely related to a class of programming problems,and matrix eigenvalue complementarity problem can be converted to the corresponding programming problem by Rayleigh quotient.Similar to the matrix eigenvalue complementarity problem,the solution of tensor eigenvalue complementarity problem can be transformed into homogeneous polynomial fractional programming problem.The equivalent transformation of the tensor generalized higher-degree eigenvalue complementarity problem is the key content of this article.In this article,we first introduce the relation between the complementarity problem and some programming problems,the method to solve the matrix eigenvalue problem using Rayleigh fraction,and the equivalent form of the matrix eigenvalue complementarity problem.Secondly,the related symbols and definitions are introduced.Furthermore,the existence of solutions to the tensor generalized eigenvalue complementarity problem and the tensor higher-degree eigenvalue complementary problem is explained.Finally,a class of equivalent transformation is proposed for the tensor generalized higher-degree eigenvalue complementarity problem.And in the case of the degree of eigenvalues satisfing k(28)2l,it is proved that the tensor generalized higher-degree eigenvalue complementarity problem can be transformed into the tensor higher-degree eigenvalue complementary problem.