Two methods for the maximization of homogeneous polynomials over the simplex

Ahmed, Faizan; Still, Georg J.

doi:10.1007/s10589-021-00307-1

Cited by 1 publication

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“…In the future, we work towards utilizing these hierarchies for providing approximation results related to copositive optimization, especially to recover approximation results for polynomial optimization over the simplex as obtained by De Klerk and co-authors [21][22][23][24]. Furthermore, our aim is to use these approximation hierarchies to develop numerical algorithms for application domains such as approximating clique numbers for uniform hypergraphs (see, e.g., [37,38]).…”

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confidence: 99%

Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex

Iqbal

Ahmed

2022

Mathematics

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In this paper, we discuss the cone of copositive tensors and its approximation. We describe some basic properties of copositive tensors and positive semidefinite tensors. Specifically, we show that a non-positive tensor (or Z-tensor) is copositive if and only if it is positive semidefinite. We also describe cone hierarchies that approximate the copositive cone. These hierarchies are based on the sum of squares conditions and the non-negativity of polynomial coefficients. We provide a compact representation for the approximation based on the non-negativity of polynomial coefficients. As an immediate consequence of this representation, we show that the approximation based on the non-negativity of polynomial coefficients is polyhedral. Furthermore, these hierarchies are used to provide approximation results for optimizing a (homogeneous) polynomial over the simplex.

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