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

Muhammad Faisal Iqbal and Faizan Ahmed
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Muhammad Faisal Iqbal: Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan
Faizan Ahmed: Formal Methods and Tools Group, University of Twente, 7522 NB Enschede, The Netherlands

Mathematics, 2022, vol. 10, issue 10, 1-17

Abstract: 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.

Keywords: copositive tensor; positive semidefinite tensor; sum of squares; approximation hierarchies; polynomial optimization; simplex (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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