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

Shenglong Hu, Guoyin Li, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

19 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)446-474
Number of pages29
JournalJournal of Optimization Theory and Applications
Issue number2
Publication statusPublished - 1 Feb 2016


  • Alternative theorem
  • Nonconvex polynomial optimization
  • Semidefinite programming
  • Sum-of-squares relaxation
  • Symmetric tensors

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

Cite this