A Semidefinite Relaxation Method for Linear and Nonlinear Complementarity Problems with Polynomials

This paper considers semidefinite relaxation for linear and nonlinear complementarity problems. For some particular copositive matrices and tensors, the existence of a solution for the corresponding complementarity problems is studied. Under a general assumption, we show that if the solution set of a complementarity problem is nonempty, then we can get a solution by the semidefinite relaxation method; while if it does not have a solution, we can obtain a certificate for the infeasibility. Some numerical examples are given.