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Theorems of the Alternative for Inequality Systems of Real Polynomials

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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.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant No. 11171252 and Grant No. 30870713). The authors are very grateful to the three referees for their valuable suggestions and constructive comments, which have considerably improved the presentation of the paper.

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  1. Sheng-Long Hu

    Present address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

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  1. Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, P.R. China

    Sheng-Long Hu & Zheng-Hai Huang

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Correspondence to Zheng-Hai Huang.

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Hu, SL., Huang, ZH. Theorems of the Alternative for Inequality Systems of Real Polynomials. J Optim Theory Appl 154, 1–16 (2012). https://doi.org/10.1007/s10957-012-9993-9

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  • DOI: https://doi.org/10.1007/s10957-012-9993-9

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