Given a set of H-infinity design specifications, the issue is to check whether there exists a controller, whose order is free, which satisfies these specifications. The classical solution, which is based on Youla parametrization and convex closed loop design, is not really satisfactory since it should use an infinite dimensional basis of filters, which cannot be done in practice. Let J* the minimal value of the design objective over such an infinite dimensional basis of. filters. A Nevanlinna Pick interpolation method is proposed here to compute lower and upper bounds of J*, by solving the design problem on a finite set of frequencies. Finite-time convergence of the algorithm is proved.