Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 158–184 | Cite as

Global Optimization of Polynomials over Real Algebraic Sets

  • Chu WangEmail author
  • Zhi-Hong Yang
  • Lihong Zhi


Let f, g1,..., gs be polynomials in R[X1,..., Xn]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum f* of f over an arbitrary given real algebraic set V = {x ∈ Rn | g1(x) = 0,..., gs(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ Rn | g1(x) ≥ 0,..., gs(x) ≥ 0}.


Polynomial optimization real algebraic set generalized critical value 


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We would like to thank Mohab Safey El Din for valuable comments and suggestions on the topic of the paper.


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Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Beijing Jinghang Computation and Communication Research InstituteBeijingChina
  2. 2.KLMM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.University of Chinese Academy of SciencesBeijingChina

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