In this paper, we describe a relaxation method to compute the minimal critical value of a real polynomial function on a semialgebraic set S and the ideal defining the points at which the minimal critical value is reached. We show that any relaxation hierarchy which is the projection of the Karush-Kuhn-Tucker relaxation stops in a finite number of steps and the ideal defining the minimizers is generated by the kernel of the associated moment matrix in that degree. Assuming the minimizer ideal is zero-dimensional, we give a new criterion to detect when the minimum is reached and we prove that this criterion is satisfied for a sufficiently high degree. This exploits new representation of positive polynomials as elements of the preordering modulo the KKT ideal, which only involves polynomials in the initial set of variables.