Skip to main content
SpringerLink
Log in

Tensor Complementarity Problems—Part III: Applications

  • Invited Paper
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We have reviewed some theoretical and algorithmic developments for tensor complementarity problems and related models in the first part and the second part of this paper, respectively. In this part, we present a survey for some applications of tensor complementarity problems and polynomial complementarity problems. We first describe some equivalent classes of tensor complementarity problems and polynomial complementarity problems, since many practical problems can be modeled as forms of those equivalent problems; and then, we review three practical applications of tensor complementarity problems and polynomial complementarity problems. The first practical application is about a class of multi-person noncooperative games, which is modeled as a tensor complementarity problem, and particularly, an explicit relationship between the solutions to these two classes of problems is presented. The second practical problem is about the hypergraph clustering problem, which can be solved by a tensor complementarity problem. The third practical problem is about a class of traffic equilibrium problems, which is modeled as a polynomial complementarity problem. Some further issues are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lemke, C.E., Howson Jr., J.T.: Equilibrium points of bimatrix games. SIAM J. Appl. Math. 12(2), 413–423 (1964)

    Article  MathSciNet  Google Scholar 

  2. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)

    Article  MathSciNet  Google Scholar 

  3. Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003)

    MATH  Google Scholar 

  4. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)

    MATH  Google Scholar 

  5. Han, J., Xiu, N., Qi, H.D.: Nonlinear Complementarity Theory and Algorithms. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)

    Google Scholar 

  6. Huang, Z.H., Qi, L.: Tensor complementarity problems—part I: basic theory. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01566-z

    Article  MathSciNet  Google Scholar 

  7. Qi, L., Huang, Z.H.: Tensor complementarity problems—part II: solution methods. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01568-x

    Article  MathSciNet  Google Scholar 

  8. Nash, J.F.: Equilibrium points in \(N\)-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)

    Article  MathSciNet  Google Scholar 

  9. Huang, Z.H., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66(3), 557–576 (2017)

    Article  MathSciNet  Google Scholar 

  10. Jain, A.K., Murty, M.N., Flynn, P.J.: Data clustering: a review. ACM Comput. Surv. 31(3), 264–323 (1999)

    Article  Google Scholar 

  11. Jain, A.K.: Data clustering: 50 years beyond \(K\)-means. Pattern Recognit. Lett. 31, 651–666 (2010)

    Article  Google Scholar 

  12. Gan, G., Ma, C., Wu, J.: Data Clustering: Theory, Algorithms, and Applications. SIAM, Philadelphia (2010)

    MATH  Google Scholar 

  13. Zien, J.Y., Schlag, M.D.F., Chan, P.K.: Multilevel spectral hypergraph partitioning with arbitrary vertex sizes. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 18(9), 1389–1399 (1999)

    Article  Google Scholar 

  14. Bolla, M.: Spectral, Euclidean representations and clusterings of hypergraphs. Discret. Math. 117, 19–39 (1993)

    Article  MathSciNet  Google Scholar 

  15. Rodríguez, J.: On the Laplacian spectrum and walk-regular hypergraphs. Linear Multilinear Algebra 51, 285–297 (2003)

    Article  MathSciNet  Google Scholar 

  16. Zhou, D., Huang, J., Schölkopf, B.: Learning with hypergraphs: clustering, classification, embedding. In: Advances in Neural Information Processing Systems, Proceedings, vol. 19, pp. 1601–1608 (2006)

  17. Bulo, S.R., Pelillo, M.: A game-theoretic approach to hypergraph clustering. IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1312–1327 (2013)

    Article  Google Scholar 

  18. Garg, V.K., Narahari, Y., Murty, M.N.: Novel biobjective clustering (bigc) based on cooperative game theory. IEEE Trans. Knowl. Data Eng. 25(5), 1070–1082 (2013)

    Article  Google Scholar 

  19. Liu, H., Latecki, L.J., Yan, S.: Revealing cluster structure of graph by path following replicator dynamic. CoRR (2013). arXiv:1303.2643

  20. Ferris, M.C., Meeraus, A., Rutherford, T.F.: Computing Wardropian equilibrium in a complementarity framework. Optim. Method Softw. 10(5), 669–685 (1999)

    Article  Google Scholar 

  21. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)

    Book  Google Scholar 

  22. Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228–238 (2013)

    Article  MathSciNet  Google Scholar 

  23. Chen, H.B., Huang, Z.H., Qi, L.: Copositivity detection of tensors: theory and algorithm. J. Optim. Theory Appl. 174, 746–761 (2017)

    Article  MathSciNet  Google Scholar 

  24. Chen, H.B., Huang, Z.H., Qi, L.: Copositive tensor detection and its applications in physics and hypergraphs. Comput. Optim. Appl. 69, 133–158 (2018)

    Article  MathSciNet  Google Scholar 

  25. Li, L., Zhang, X., Huang, Z.H., Qi, L.: Test of copositive tensors. J. Ind. Manag. Optim. 15(2), 881–891 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Li, X., Ng, M.K.: Solving sparse non-negative tensor equations: algorithms and applications. Front. Math. China 10(3), 649–680 (2015)

    Article  MathSciNet  Google Scholar 

  27. Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \(Z\)-tensor complementarity problems. Optim. Lett. 11(3), 471–482 (2017)

    Article  MathSciNet  Google Scholar 

  28. Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73(4), 789–800 (2019)

    Article  MathSciNet  Google Scholar 

  29. Isac, G.: Topological Methods in Complementarity Theory. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  30. Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceeding of the Institute of Civil Engineers, Part II, pp 325–378 (1952)

  31. Aashtiani, H.Z., Magnanti, T.L.: Equilibria on a congested transportation network. SIAM J. Algebr. Discret. Method 2(3), 213–226 (1981)

    Article  MathSciNet  Google Scholar 

  32. Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11, 681–689 (1965)

    Article  MathSciNet  Google Scholar 

  33. Hansen, T., Manne, A.S.: Equilibrium and linear complementarity—an economy with institutional constraints on prices. In: Schwödiauer, G. (ed.) Equilibrium and Disequilibrium in Economic Theory, pp. 227–237. D. Reidel, Dordrecht (1978)

    Google Scholar 

  34. Maier, G., Nonati, G.: Elastic-plastic boundary element analysis as a linear complementarity problem. Appl. Math. Model. 7(2), 74–82 (1983)

    Article  MathSciNet  Google Scholar 

  35. Bolzon, G.: Complementarity problems in structural engineering: an overview. Arch. Comput. Methods Eng. 24, 23–36 (2017)

    Article  MathSciNet  Google Scholar 

  36. Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley Series in Nonlinear Science. Wiley, New York (1996)

    Book  Google Scholar 

  37. van Eijndhoven, J.T.J.: Solving the linear complementarity problem in circuit simulation. SIAM J. Control Optim. 24, 1050–1062 (1986)

    Article  MathSciNet  Google Scholar 

  38. Cryer, C.W., Lin, Y.: An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary problems. Appl. Math. Optim. 13, 1–17 (1985)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very grateful to professors Chen Ling, Yisheng Song, Shenglong Hu, and Ziyan Luo for reading the first draft of this paper and putting forward valuable suggestions for revision. The first author’s work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002 and 11871051), and the second author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716, and 15300717).

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Zheng-Hai Huang

  2. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Liqun Qi

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi

Authors
  1. Zheng-Hai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liqun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqun Qi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, ZH., Qi, L. Tensor Complementarity Problems—Part III: Applications. J Optim Theory Appl 183, 771–791 (2019). https://doi.org/10.1007/s10957-019-01573-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01573-0

Keywords

Mathematics Subject Classification

Search

Navigation