TY - JOUR
T1 - Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex
AU - Iqbal, Muhammad Faisal
AU - Ahmed, Faizan
N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.
Financial transaction number:
6100001268
PY - 2022/5/14
Y1 - 2022/5/14
N2 - 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.
AB - 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.
KW - approximation hierarchies
KW - copositive tensor
KW - polynomial optimization
KW - positive semidefinite tensor
KW - simplex
KW - sum of squares
UR - http://www.scopus.com/inward/record.url?scp=85130546551&partnerID=8YFLogxK
U2 - 10.3390/math10101683
DO - 10.3390/math10101683
M3 - Article
AN - SCOPUS:85130546551
SN - 2227-7390
VL - 10
JO - Mathematics
JF - Mathematics
IS - 10
M1 - 1683
ER -