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A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure

Author

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  • Shenglong Hu

    (Tianjin University)

  • Guoyin Li

    (University of New South Wales)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

Abstract

Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially nonpositive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:2:d:10.1007_s10957-014-0652-1
    DOI: 10.1007/s10957-014-0652-1
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    References listed on IDEAS

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    1. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, December.
    2. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    3. Shenglong Hu & Liqun Qi, 2012. "Algebraic connectivity of an even uniform hypergraph," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 564-579, November.
    4. Shenglong Hu & Guoyin Li & Liqun Qi & Yisheng Song, 2013. "Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 717-738, September.
    5. Liqun Qi & Yinyu Ye, 2014. "Space tensor conic programming," Computational Optimization and Applications, Springer, vol. 59(1), pages 307-319, October.
    6. Immanuel Bomze & Chen Ling & Liqun Qi & Xinzhen Zhang, 2012. "Standard bi-quadratic optimization problems and unconstrained polynomial reformulations," Journal of Global Optimization, Springer, vol. 52(4), pages 663-687, April.
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    Cited by:

    1. T. D. Chuong & V. Jeyakumar & G. Li, 2019. "A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs," Journal of Global Optimization, Springer, vol. 75(4), pages 885-919, December.
    2. Meijia Yang & Shu Wang & Yong Xia, 2022. "Toward Nonquadratic S-Lemma: New Theory and Application in Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 353-363, July.
    3. 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.
    4. Qingzhi Yang & Yang Zhou & Yuning Yang, 2019. "An Extension of Yuan’s Lemma to Fourth-Order Tensor System," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 803-810, March.

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