Fast Algorithms For Tensor Eigenvalue Complementarity Problems

Tensor is a generalization of matrix,however,the tensor-related problems are more difficult to be solved than matrix-related ones because the complicated structure of tensor.The matrix eigenvalue complementarity problem which has many applications in mechanics is a special kind of nonlinear complementarity problems.Tensor eigenvalue complementarity problem is a high-ordcr generalization of matrix eigenvalue complementarity problem,which is widely used in en-gineering field.In recent years,tensor eigenvalue complementarity problem has received much considerable attention in the optimization community.The problem is challenging due to the high nonlinearity of the tensor eigenvalue complementarity problem.To our knowledge,there are few algorithms for solving tensor eigenvalue complementarity problems,and most solvers are focused on the low-dimensional tensor eigenvalue complementarity problems.Base on the exist-ing research results,we reformulate tensor eigenvalue complementarity problem as the Rayleigh quotient maximization problem over the simplex constraint.Three algorithms are introduced to find Pareto-eigenvalues of tensor eigenvalue complementarity problems.In Chapter 3,we propose an improved spectral projected gradient algorithm,the key idea of the algorithm is to adopt the absolute value of BB step size,which can make full use of the advantages of the BB step size.Numerical experiments demonstrate that the improved spectral projected gradient algorithm works well in terms of taking less iteration and computing time than the existing spectral projected gradient algorithms.In Chapter 4,we propose a non-monotone spectral projected gradient algorithm for solv-ing high-order and high-dimensional tensor eigenvalue complementarity problems.As the di-mension of the problem increases,the difficulty of solving the tensor eigenvalue complemen-tarity problem further increases.Therefore,we combine absolute value of BB step size and non-monotone linesearch technique to propose non-monotone spectral projected gradient algo-rithm.The algorithm reduces the amount of inner iterations,thereby reducing much computing time.Under certain assumptions,the sublinear convergence rate of the non-monotone spec-tral projected gradient algorithm is analyzed.Numerical experiments demonstrate that the non-monotone spectral projected gradient algorithm performs better than the existing algorithms in solving the high-dimensional tensor eigenvalue complementarity problem.Moreover,our method is insensitive on the choice of initial points.In Chapter 5,we propose an alternating direction method of multipliers to solve tensor eigen-value complementarity problems.The iterative scheme of the alternating direction method of multipliers is very simple,but one of the two sub-problems in iteration has no closed-form solu-tion.Therefore,two strategies for solving sub-problem are proposed:iterative method and prox-imal point approximation.Finally,numerical experiments are performed to compare alternating direction method of multipliers with the non-monotone spectral projected gradient algorithm and the spectral projected algorithm.The results show that although the alternating direction method of multipliers requires more iterations in numerous examples but less computing time than the spectral projected gradient algorithm.In addition,the non-monotone spectral projected gradient algorithm works well in terms of taking less iteration and computing time.