Abstract
A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Bulo and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local (global) minimizers of a homogeneous polynomial function of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Turan densities of complete non-uniform hypergraphs.
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