One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality obtained by a Positivstellensatz. This problem itself reduces to a feasibility problem in semidefinite programming. Unfortunately, this last problem is in general not strictly feasible --- the solution set then is a convex with empty interior, which precludes direct use of numerical optimization methods. We propose a workaround for this difficulty. We implemented our method and illustrate its use with examples including automatically showing that various nonnegative polynomials that have been shown not to be sums of squares of polynomials are indeed nonnegative, as quotients of two sums of squares of polynomials.