  Motzkin and Straus established a remarkable connection between the maximum
clique and the Lagrangian of a graph in 1965. This connection and its
extensions were successfully employed in optimization to provide heuristics for
the maximum clique number in graphs. It is useful in practice if similar
results hold for hypergraphs. In this paper, we provide upper bounds on the
Lagrangian of a hypergraph containing dense subgraphs when the number of edges
of the hypergraph is in certain ranges. These results support a pair of
conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of J.
Talbot (2002). \keywords{Cliques of hypergraphs \and Colex ordering \and
Lagrangians of hypergraphs \and Polynomial optimization}
