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Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities

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Abstract

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as a polynomial optimization problem by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization problem. We also prove that the assumption of nonsingularity in Nie’s method can be weakened to the finiteness of singularities. Some numerical examples are given in the end.

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Acknowledgments

The research was partially supported by NSF grant DMS-0844775 and Research Fund for Doctoral Program of Higher Education of China grant 20114301120001. The authors thank the referees for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Feng Guo & Li Wang

  2. School of Mathematics and Computing Science, Xiangtan University, Hunan, 411105, People’s Republic of China

    Guangming Zhou

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  1. Feng Guo
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  2. Li Wang
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  3. Guangming Zhou
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Correspondence to Feng Guo.

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Guo, F., Wang, L. & Zhou, G. Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities. J Glob Optim 58, 261–284 (2014). https://doi.org/10.1007/s10898-013-0047-0

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