A mixed integer programming approach to the tensor complementarity problem

In this paper, we establish a new approach to solve the tensor complementarity problem (TCP). A mixed integer programming model is given and the TCP is solved by solving the model. The TCP is shown to be formulated as an equivalent mixed integer feasibility problem. Based on the reformulation, some conditions are obtained to guarantee the solvability of the TCP. Specially, a sufficient condition is given for TCP without solutions. A necessary and sufficient condition is given for existence of solutions. We also give a concrete bound for the solution set of the TCP with positive definite tensors. Moreover, we show that the TCP with a diagonal positive definite tensor has a unique solution. Numerical experiments on several test problems illustrate the efficiency of the proposed approach in terms of the quality of the obtained solutions.