A Class Of Second-order Cone Eigenvalue Complementarity Problems For Higher-order Tensors

This thesis focuses on the second-order cone tensor eigenvalue complementarity problems(SOCTEiCP)theoretical analysis and algorithm design.We first show the equivalence of the SOCTEiCP to an appropriate nonlinear programming problems.Furthermore,a necessary and sufficient condition for the stationary point of the corresponding nonlinear programming problem being the solution of the SOCTEiCP is given.An implementable projection method is also developed for the SOCTEiCP under consideration.Matrix eigenvalue complementarity problem is a special formation of the complementarity problem which has many widely applications.Eigenvalue complementarity problem of tensors,which is a natural extension of matrix eigenvalue complementarity problem has closely connection with a class of nonlinear differential inclusion problems.It is clear that the SOCTEiCP is not only a natural extension of the second-order cone matrix eigenvalue complementarity problem,but also a type of special complementarity problem of tensors.The solving of tensor eigenvalue complementarity problem is an NPhard problem.In general,we transform the eigenvalue complementarity problem into an equivalent nonlinear programming problem.In this paper,we show the equivalence of the SOCTEiCP to a particular variational inequality,thereby giving a positive answer that the SOCTEiCP has at least one solution.And we also introduce the equivalent relation between the symmetric SOCTEiCP and nonlinear programming.Furthermore,a necessary and sufficient condition for the stationary point of the nonlinear programming problem being the solution of the SOCTEiCP is given.In addition,we develop a projection algorithm to solve the symmetric SOCTEiCP and report some preliminary numerical results to verify the reliability of the proposed algorithm.As a more general case,we discuss the case of the sub-symmetric SOCTEiCP,and characterize the solution of the problem,which shows that the solution of the problem can be transformed into the stationary point of a class of nonlinear optimization problem under certain conditions.The main contents are as follows: First of all,review the development situations of complementarity problems,matrix eigenvalue complementarity problems and tensor eigenvalue complementarity problems.Secondly,we show that SOCTEiCP is provably equivalent to a variational inequality,thereby establishing the existence of a solution to SOCTEiCP.Again,we consider the symmetric SOCTEiCP.Based upon such a symmetry condition,we can gainfully formulate the symmetric SOCTEiCP as a fractional polynomial optimization problem,which help us develop a projection-based method for finding its solutions.Finally,as a more general case,we discuss the sub-symmetric SOCTEiCP.Similarly,we also give a nonlinear programming formulation for the sub-symmetric SOCTEiCP.Based upon this,we get the existence of the solution of the SOCTEiCP.