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Univariate Representations of Solutions to Generic Polynomial Complementarity Problems

Author

Listed:
  • Vu Trung Hieu

    (Center for Advanced Intelligence Project, RIKEN)

  • Alfredo Noel Iusem

    (Fundação Getulio Vargas)

  • Paul Hugo Schmölling

    (Norwegian University of Science and Technology)

  • Akiko Takeda

    (Center for Advanced Intelligence Project, RIKEN
    The University of Tokyo)

Abstract

By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper points out that, generically, this polynomial system has finitely many complex zeroes. In such a case, we use symbolic computation techniques to compute a univariate representation of the solution set. Consequently, univariate representations of special solutions, such as least-norm and sparse solutions, are obtained. After that, enumerating solutions boils down to solving problems governed by univariate polynomials. We also provide some experiments on small-scale problems with worst-case scenarios. At the end of the paper, we propose a method for computing approximate solutions to copositive polynomial complementarity problems that may have infinitely many solutions.

Suggested Citation

  • Vu Trung Hieu & Alfredo Noel Iusem & Paul Hugo Schmölling & Akiko Takeda, 2025. "Univariate Representations of Solutions to Generic Polynomial Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 207(2), pages 1-22, November.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:2:d:10.1007_s10957-025-02788-0
    DOI: 10.1007/s10957-025-02788-0
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    References listed on IDEAS

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    1. Lixing Han, 2019. "A Continuation Method for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 949-963, March.
    2. Yang Xu & Guyan Ni & Mengshi Zhang, 2024. "Bounds of the Solution Set to the Polynomial Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 146-164, October.
    3. Zheng-Hai Huang & Liqun Qi, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    4. Vu Trung Hieu, 2020. "Solution maps of polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 77(4), pages 807-824, August.
    5. Vu Trung Hieu & Akiko Takeda, 2025. "Computing local minimizers in polynomial optimization under genericity conditions," Journal of Global Optimization, Springer, vol. 92(4), pages 909-932, August.
    6. Yisheng Song & Liqun Qi, 2016. "Tensor Complementarity Problem and Semi-positive Tensors," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1069-1078, June.
    7. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 1-23, October.
    8. Maolin Che & Liqun Qi & Yimin Wei, 2016. "Positive-Definite Tensors to Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 475-487, February.
    9. Vu Trung Hieu, 2019. "On the R0-Tensors and the Solution Map of Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 163-183, April.
    10. Shouqiang Du & Liping Zhang, 2019. "A mixed integer programming approach to the tensor complementarity problem," Journal of Global Optimization, Springer, vol. 73(4), pages 789-800, April.
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