In 1965, Motzkin and Straus provided a connection between the order of a maximum clique in a graph G and a global solution of a quadratic optimization problem determined by G which is called the Lagrangian function of G. This connection and its extensions have been useful in both combinatorics and optimization. In 2009, Rota Bulò and Pelillo extended the Motzkin-Straus result to r -uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree r (different from Lagrangian function) and the maximal (maximum) cliques of an r -uniform hypergraph. In this paper, we study similar optimization problems related to nonuniform hypergraphs and obtain some extensions of their results to non-uniform hypergraphs. In particular, we provide a one-to-one correspondence between local (global) minimizers of a family of non-homogeneous polynomial functions and the maximal (maximum) cliques of {1, r }-hypergraphs. An application of a main result gives an upper bound on the Turán density of complete {1, r }-hypergraphs. The approach applied in the proof follows from the approach in Rota Bulò and Pelillo (2009).
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