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

Shenglong Hu Tianjin University Guoyin Li University of New South Wales Liqun Qi Hong Kong Polytechnic University

Optimization and Control mathscidoc:1904.27004

Journal of Optimization Theory and Applications, 168, (2), 446–474, 2016
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.
Tensor, polynomial optimization, theorem of the alternative
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@inproceedings{shenglong2016a,
  title={A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure},
  author={Shenglong Hu, Guoyin Li, and Liqun Qi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190428220941193252289},
  booktitle={Journal of Optimization Theory and Applications},
  volume={168},
  number={2},
  pages={446–474},
  year={2016},
}
Shenglong Hu, Guoyin Li, and Liqun Qi. A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure. 2016. Vol. 168. In Journal of Optimization Theory and Applications. pp.446–474. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190428220941193252289.
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