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

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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)

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.

Suggested Citation

  • Muhammad Faisal Iqbal & Faizan Ahmed, 2022. "Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex," Mathematics, MDPI, vol. 10(10), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1683-:d:815574
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    References listed on IDEAS

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    1. Shenglong Hu & Guoyin Li & Liqun Qi, 2016. "A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 446-474, February.
    2. de Klerk, E. & Laurent, M. & Parrilo, P., 2006. "A PTAS for the minimization of polynomials of fixed degree over the simplex," Other publications TiSEM 603897c9-179e-43e4-9e83-6, Tilburg University, School of Economics and Management.
    3. Olga Kostyukova & Tatiana Tchemisova, 2021. "Structural Properties of Faces of the Cone of Copositive Matrices," Mathematics, MDPI, vol. 9(21), pages 1-21, October.
    4. Tareq Hamadneh & Mohammed Ali & Hassan AL-Zoubi, 2020. "Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    5. Faizan Ahmed & Georg Still, 2021. "Two methods for the maximization of homogeneous polynomials over the simplex," Computational Optimization and Applications, Springer, vol. 80(2), pages 523-548, November.
    6. NESTEROV, Yurii, 1999. "Global quadratic optimization on the sets with simplex structure," LIDAM Discussion Papers CORE 1999015, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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