Newton Type Algorithms For Tensor Complementarity Problems And Tensor Eigenvalue Complementarity Problems

Tensor complementarity problem(TCP)and tensor eigenvalue complementarity problem(TEiCP)are among the research hotspots of tensor optimization field,and have important application in game theory and engineering calculation.In this paper,we propose semismooth Newton method for TCP and smoothing Newton method for TEiCP,analyze the convergence and show the numerical results.The main results are as follows:(1).By introduction a new kind of NCP function,we transform TCP into nonsmooth equations,and prove its semismoothness.We design a Newton method to solve the equations by combining a new calculation method for certain subdifferential,achieve some interesting properties and prove that the algorithm has local quadratic convergence under some conditions.Finally,we use the semismooth Newton algorithm to calculate the Nash equilibrium of n-person noncooperative game.The numerical results show that the algorithm is valid.(2).We design a smoohting Newton method by introduction a special kind of smoothing function named CHKS function.We prove that under some assumptions any accumulation point generated by the algorithm is the solution of TEiCP.Especially,we conduct numerical experiments to compare our algorithm with other existing algorithms.The numerical results show that our algorithm is effective and have some advantage than other algorithms.