The Fast Iterative Algorithms For Some Tensor Complementarity Poblems

As a special case of nonlinear complementarity problems,the tensor complementarity problem is a hot research topic in recent years.Many practical models can be described by the stochastic tensor complementarity problems.The modulus-based matrix splitting iteration method is a classical method for solving the complementary problems.In this paper,some fast iterative algorithms for solving the tensor complementarity problems and the stochastic tensor complementarity problems are constructed by incorporating the modulus-based matrix splitting iteration methods and tensor splitting.The details are as follows:Firstly,a class of smoothing modulus-based matrix iteration method is proposed to solve the tensor complementarity problems.The basic idea is that,we firstly transform the tensor complementarity problem into an equivalent system of modulus-based equations,then introduce an approximated smoothing function to obtain its approximation solutions,and the convergence is analyzed.The efficiency of the algorithm is verified by some numerical experiments.Secondly,based on the tensor splitting,we establish a class of lower dimensional equations method for solving the tensor complementarity problems.Different types of iterative methods are obtained by different splittings of the coefficient tensors in the lower dimensional tensor equations.The convergence analysis and numerical experimental results of the algorithm are given.Thirdly,we give a definition of stochastic strongly strictly semi-positive tensor,and transform the stochastic tensor complementarity problems into the tensor complementarity problems by empolying the expected value formulation.The regularized smoothing modulus-based matrix iteration method is derived.Numerical examples are given rise to show the effectiveness of the algorithm.