The Symmetric Tensor Eigenvalue Complementarity Problem

Complementarity problem plays an important role in mathematical programming,and is closely related to optimization problem,variational principle and fixed point theory.The tensor eigenvalue complementarity problem is a generalization of the complementarity problem and the eigenvalue problem.In this paper,we mainly study the even order symmetric tensor eigen-value complementarity problem(OTEiCP)and the symmetric tensor higher-degree eigenvalue complementarity problem(OTHEiCP).This paper first introduces the tensor eigenvalue problem(TEiP),and give a necessary and sufficient condition to determine the positive definiteness(or positive semidefiniteness)of ten-sor.The generalized tensor eigenvalue problem(GTEiP)is an extension of TEiP.We give the definition of generalized Rayleigh quotient function and point out that the stationary points of generalized Rayleigh quotient function are the solutions of the symmetric generalized tensor eigenvalue problem.The symmetric tensor eigenvalue complementarity problem can be transformed into finding a stationary point of the Rayleigh quotient function on the simplex.Furthermore,the OTEiCP is shown to be equivalent to three nonlinear programmings(NLP).The merit functions of NLP are Rayleigh quotient,logarithm and polynomial functions,respectively.In view of the above three nonlinear programmings,we give the corresponding propositions and theorems.In fact,every stationary point x of NLP with A(x)>0 is a solution of the OTEiCP and the optimal solution x of NLP with A(x)>0 is the maximum A-solution of the OTEiCP.This paper also points out that OTEiCP is solvable if and only if satisfying Axm>0 for some x? 0,and proves the solu-tion's existence of OTEiCP under A taking some special tensors.Then,we propose a multiplier method for OTEiCP.The numerical experiments also show that the validity and reliability of the algorithm.On the basis of the symmetric tensor eigenvalue complementarity problem,we also explore OTHEiCP.Due to the complexity of the symmetric tensor higher-degree eigenvalue complemen-tarity problem,the existence conditions of solutions are more stringent.In this paper,two special cases of OTHEiCP are studied:when m?2,OTHEiCP is transformed into the quadratic eigen-value complementary problem(QEiCP).In the case of A?O,B is a positive definite tensor or B?O,C is a positive definite tensor,OTHEiCP can be transformed into the symmetric tensor eigenvalue complementarity problem.At the same time,the existence theorem of the above OTHEiCP solutions is given.