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Notes on the Optimization Problems Corresponding to Polynomial Complementarity Problems

  • Vu Trung HieuEmail author
  • Yimin Wei
  • Jen-Chih Yao
Article
  • 59 Downloads

Abstract

This work is motivated by a conjecture of Che et al. (J Optim Theory Appl 168:475–487, 2016) which says that if the feasible region of a tensor complementarity problem is nonempty, then the corresponding optimization problem has a solution. The aim of the paper is twofold. First, we show several sufficient conditions for the solution existence of the optimization problems corresponding to polynomial complementarity problems. Consequently, some results for tensor complementarity problems are obtained. Second, we disprove the conjecture by giving a counterexample.

Keywords

Polynomial complementarity problem Tensor complementarity problem Polynomial optimization problem Feasible region Solution existence 

Mathematics Subject Classification

90C33 11C08 

Notes

Acknowledgements

The authors are grateful to the editor-in-chief and the anonymous referees for their valuable suggestions. The first author wishes to thank the Department of Applied Mathematics, National Sun Yat-Sen University and the Center for General Education, China Medical University for hospitality and support. The research of Yimin Wei is supported by the National Natural Science Foundation of China under Grant 11771099 and the Innovation Program of Shanghai Municipal Education Committee.

References

  1. 1.
    Gowda, M.S.: Polynomial complementarity problems. Pac. J. Optim. 13, 227–241 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ling, L., He, H., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67, 341–358 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Wang, J., Hu, S., Huang, Z.H.: Solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 176, 120–136 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hu, S., Wang, J., Huang, Z.H.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 179, 983–1000 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)zbMATHGoogle Scholar
  6. 6.
    Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)CrossRefGoogle Scholar
  8. 8.
    Huang, Z.H., Qi, L.: Tensor complementarity problems—part I: basic theory. J. Optim. Theory Appl. 183, 1–23 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Qi, L., Huang, Z.H.: Tensor complementarity problems—part II: solution methods. J. Optim. Theory Appl. 183, 365–385 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Huang, Z.H., Qi, L.: Tensor complementarity problems—part III: applications. J. Optim. Theory Appl. (2019).  https://doi.org/10.1007/s10957-019-01573-0 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klatte, D.: On a Frank–Wolfe type theorem in cubic optimization. Optimization 68, 539–547 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Obuchowska, W.T.: On generalizations of the Frank–Wolfe theorem to convex and quasi-convex programmes. Comput. Optim. Appl. 33, 349–364 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hieu, V.T.: A regularity condition in polynomial optimization. arXiv:1808.06100

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of MathematicsPhuong Dong UniversityHanoiVietnam
  2. 2.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan

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