Quasi-Newton Method Of Tensor Equation And Complementarity Problem

This paper mainly studies the structured quasi-Newton algorithm of M tensor equations and M tensor complementarity problems.Tensor equations and tensor complementarity problems have a wide range of practical applications.The study of numerical algorithms for tensor equations and complementary problems is also a topic of common concern in the optimization community and the numerical algebra community in recent years.First,based on the approximate Newton method for solving tensor equations,this paper uses the special structure of the Jacobi matrix of tensor equations to propose a structured quasi-Newton method for solving M tensor equations.The advantage of this algorithm is that the quasi-Newton matrix generated by the algorithm is an M-matrix,and part of the Jacobi matrix information is retained.If the initial point is selected appropriately,then the sequence of points generated by the algorithm monotonically converges to a non-negative solution of the M tensor equation.We test the proposed algorithm through numerical experiments,and compare the numerical results with the existing approximate Newton method.The results show the effectiveness of the structured quasi-Newton method proposed in this paper.On this basis,we further study the numerical method for solving the Mtensor complementarity problem.Similar to the idea of solving tensor equations,using the special structure of the Jacobi matrix of tensor equations,a structured quasi-Newton method for solving the M-tensor complementarity problem is proposed.The monotonic convergence of the algorithm is proved,and the proposed algorithm is tested through numerical experiments.The results show that the algorithm in this paper has good numerical effects.