Should a Cluster Validity Index contain the same measure(s) as the Clustering Algorithm?
It's standard to use, e.g., Silhouette to assess the quality of clusterings, that were obtained with other methods.

I'm not a big fan of this (people always think they would get the "optimal" result this way, but they don't), but it's all over literature. Logically, you are trying to maximizing Silhouette and use the other algorithms as sampling procedure for possible results.