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Theorems of the Alternative for Inequality Systems of Real Polynomials

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  • Sheng-Long Hu

    (Tianjin University
    The Hong Kong Polytechnic University)

  • Zheng-Hai Huang

    (Tianjin University)

Abstract

In this paper, we establish theorems of the alternative for inequality systems of real polynomials. For the real quadratic inequality system, we present two new results on the matrix decomposition, by which we establish two theorems of the alternative for the inequality system of three quadratic polynomials under an assumption that at least one of the involved forms be negative semidefinite. We also extend a theorem of the alternative to the case with a regular cone. For the inequality system of higher degree real polynomials, defined by even order tensors, a theorem of the alternative for the inequality system of two higher degree polynomials is established under suitable assumptions. As a byproduct, we give an equivalence result between two statements involving two higher degree polynomials. Based on this result, we investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions.

Suggested Citation

  • Sheng-Long Hu & Zheng-Hai Huang, 2012. "Theorems of the Alternative for Inequality Systems of Real Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 1-16, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-012-9993-9
    DOI: 10.1007/s10957-012-9993-9
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    References listed on IDEAS

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    1. Wenbao Ai & Yongwei Huang & Shuzhong Zhang, 2008. "On the Low Rank Solutions for Linear Matrix Inequalities," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 965-975, November.
    2. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    3. 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.
    4. Yongwei Huang & Shuzhong Zhang, 2007. "Complex Matrix Decomposition and Quadratic Programming," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 758-768, August.
    5. Anthony Man-Cho So & Yinyu Ye & Jiawei Zhang, 2008. "A Unified Theorem on SDP Rank Reduction," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 910-920, November.
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    Cited by:

    1. M. Ruiz Galán, 2017. "A theorem of the alternative with an arbitrary number of inequalities and quadratic programming," Journal of Global Optimization, Springer, vol. 69(2), pages 427-442, October.
    2. 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.
    3. 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.

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