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Global Optimization of Polynomials over Real Algebraic Sets

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Abstract

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}.

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Acknowledgement

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

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Authors and Affiliations

  1. Beijing Jinghang Computation and Communication Research Institute, Beijing, 100074, China

    Chu Wang

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhi-Hong Yang & Lihong Zhi

  3. University of Chinese Academy of Sciences, Beijing, 100049, China

    Zhi-Hong Yang & Lihong Zhi

Authors
  1. Chu Wang
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  2. Zhi-Hong Yang
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  3. Lihong Zhi
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Corresponding author

Correspondence to Chu Wang.

Additional information

The research is supported by the National Key Research Project of China under Grant No. 2018YFA0306702 and the National Natural Science Foundation of China under Grant No. 11571350.

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Wang, C., Yang, ZH. & Zhi, L. Global Optimization of Polynomials over Real Algebraic Sets. J Syst Sci Complex 32, 158–184 (2019). https://doi.org/10.1007/s11424-019-8351-5

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