On a version of cluster sampling and its practical use

If a candidate c-optimal design gives a singular information matrix, an equivalence theorem from Silvey (1978) can be used to verify optimality. The theorem is difficult to use however, as it requires a generalized inverse of the information matrix but not all generalized inverses can be used. Silvey identified finding such an inverse as an open problem. A characterization of all generalized inverses that can be used is presented here, building on Elfvingʹs (1952) geometric method for finding c-optimal designs and using the Elfving set.