Equality Based Contraction of Semidefinite Programming Relaxations in Polynomial Optimization

The SDP (semidefinite programming) relaxation for general POPs (polynomial optimization problems), which was proposed as a method for computing global optimal solutions of POPs by Lasserre, has become an active research subject recently. We propose a new heuristic method exploiting the equality constraints in a given POP, and strengthen the SDP relaxation so as to achieve faster convergence to the global optimum of the POP. We can apply this method to both of the dense SDP relaxation which was originally proposed by Lasserre, and the sparse SDP relaxation which was later proposed by Kim, Kojima, Muramatsu and Waki. Especially, our heuristic method incorporated into the sparse SDP relaxation method has shown a promising performance in numerical experiments on large scale sparse POPs. Roughly speaking, we induce valid equality constraints from the original equality constraints of the POP, and then use them to convert the dense or sparse SDP relaxation into a new stronger SDP relaxation. Our method is enlightened by some strong theoretical results on the convergence of SDP relaxations for POPs with equality constraints provided by Lasserre , Parrilo and Laurent, but we place the main emphasis on the practical aspect to compute more accurate lower bounds of larger sparse POPs.