Research On The Algorithms For Tensor Complementarity Problems

Tensor complementarity problem(TCP)is an important branch of tensor optimiza-tion,which has many applications in multi-person noncooperative games,hypergraph clustering problems,and traffic equilibrium problems.The theoretical research of TCP is very active,and it has made a lot of results in the aspects of the existence or uniqueness of solutions and the boundedness of the solution set.But there are few algorithms to solve TCP.The intrinsic function of TCP is a homogeneous polynomial based on tensor.How to design an effective numerical algorithm to solve tensor complementarity problem by analyzing its properties is an important research topic,which has important theoretical and practical significance.This thesis is devoted to the study of the algorithm for tensor complementarity problem.The main results are as follows:1.Based on the intrinsic function of TCP is monotone mapping,we analyze the existence theory of solutions of the corresponding TCP.Under the assumption that the tensor is S-tensor,we prove that the solution set is nonempty and bounded.If the intrinsic function is strictly monotone,then the corresponding TCP has a unique solution.These results are better than the general nonlinear complementarity problem.2.We design the alternating direction method of multipliers to solve TCP when the intrinsic function is monotone mapping,and prove that the algorithm is linear conver-gence.Numerical experiments show that the algorithm is effective.We also design the smoothing Newton method to solve TCP when the tensor is₀tensor.Numerical exper-iments also show the effectiveness of the algorithm.