An extension of the elimination method for a sparse sos polynomial

We propose a method to reduce the sizes of SDP relaxation problems fur a given polynomial optimization problem (POP). This method is an extension of the elimination method for a sparse SOS polynomial in [8] and exploits sparsity of polynomials involved in a given POP. In addition, we show that this method is a partial application of a facial reduction algorithm, which generates a smaller SDP problem with an interior feasible solution. In general, SDP relaxation problems for POPs often become highly degenerate because of a lack of interior feasible solutions. As a result, the resulting SDP relaxation problems obtained by this method may have an interior feasible solution, and one may be able to solve the SDP relaxation problems effectively Numerical results in this paper show that the resulting SDP relaxation problems obtained by this method can be solved fast and accurately.