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Connection Between Polynomial Optimization and Maximum Cliques of Non-Uniform Hypergraphs

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Abstract

In Motzkin and Straus (Canad. J. Math 498 17, 533540 1965) provided a connection between the order of a maximum clique in a graph G and the Lagrangian function of G. In Rota Bulò and Pelillo (Optim. Lett. 500 3, 287295 2009) 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 and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study similar optimization problems and obtain the connection to maximum cliques for {s, r}-hypergraphs and {p, s, r}-hypergraphs, which can be applied to obtain upper bounds on the Turán densities of the complete {s, r}-hypergraphs and {p, s, r}-hypergraphs.

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Acknowledgements

Supported in part by National Natural Science Foundation of China (No. 11671124).

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Authors and Affiliations

  1. College of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Pingge Chen

  2. Institute of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Yuejian Peng

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  1. Pingge Chen
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  2. Yuejian Peng
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Correspondence to Yuejian Peng.

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Chen, P., Peng, Y. Connection Between Polynomial Optimization and Maximum Cliques of Non-Uniform Hypergraphs. Order 35, 301–319 (2018). https://doi.org/10.1007/s11083-017-9434-3

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  • DOI: https://doi.org/10.1007/s11083-017-9434-3

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