An alternating direction method of multipliers for tensor complementarity problems

The tensor complementarity problem (TCP) is a special instance of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems, and traffic equilibrium problems. How to solve the TCP, via analyzing the structure of the related tensor, is one of important research issues. In this paper, we propose an alternating direction method of multipliers (ADMM) to solve the TCP. We show that the solution set of the TCP, where the involved multilinear mapping is monotone, is nonempty and compact if the involved tensor is an S-tensor. Moreover, the ADMM for the TCP with a monotone involved multilinear mapping is proven to be globally convergent with a linear convergence rate. Some preliminary numerical results show that the proposed ADMM method is promising and effective.