Connection between a class of polynomial optimization problems and maximum cliques of non-uniform hypergraphs

Abstract

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 non-uniform 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).