Skip to main content
SpringerLink
Log in

Generalizations of \(R_0\) and \(\textbf{SSM}\) Properties for Extended Horizontal Linear Complementarity Problem

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

Abstract

In the setting of the Extended Horizontal Linear Complementarity Problem, we study boundedness, nonemptyness, uniqueness and connectedness issues of solution sets. We introduce column \(R_0\)-W and column \(\textbf{SSM}\)-W properties and, using degree theory, establish boundedness and existence results. Some uniqueness results are also covered. We provide a necessary condition and a sufficient condition for EHLCP to have connected solution sets.

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. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer-Verlag, Berlin (2006)

    Book  MATH  Google Scholar 

  2. Camlibel, M.K., Schumacher, J.M.: Existence and uniqueness of solutions for a class of piecewise linear dynamical systems. Linear Algebra Appl. 351–352, 147–184 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chi, X., Gowda, M.S., Tao, J.: The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J. Glob. Optim. 73, 153–169 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cottle, R.W., Dantzig, G.B.: A generalization of the linear complementarity problem. J. Comb. Theory 8(1), 79–90 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)

    MATH  Google Scholar 

  6. De Schutter, B., De Moor, B.: The extended linear complementarity problem. Math. Program. 71(3), 289–325 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Facchinei, F., Pang, J.-S.: Finite-Dimemsional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    MATH  Google Scholar 

  8. Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math. Oper. Res. 18, 868–879 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gowda, M.S.: Reducing a monotone horizontal LCP to an LCP. Appl. Math. Lett. 8(1), 97–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  11. Jones, C., Gowda, M.S.: On the connectedness of solution sets in linear complementarity problems. Linear Algebra Appl. 272(1–3), 33–44 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kaneko, I.: A linear complementarity problem with an n by 2n “\(P\)"-matrix. In: Balinski, M.L., Cottle, R.W. (eds.) Complementarity and Fixed Point Problems Mathematical Programming Studies, vol. 7. Springer, Berlin (1978)

    Google Scholar 

  13. Mangasarian, O.L., Pang, J.-S.: The extended linear complementarity problem. SIAM J. Matrix Anal. Appl. 16(2), 359–368 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mezzadri, F., Galligani, E.: A generalization of irreducibility and diagonal dominance with applications to horizontal and vertical linear complementarity problems. Linear Algebra Appl. 621, 214–234 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mezzadri, F., Galligani, E.: Splitting methods for a class of horizontal linear complementarity problems. J. Optim. Theory Appl. 180, 500–517 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Murthy, G.S.R., Parthasarathy, T., Sriparna, B.: On the solution sets of linear complementarity problems. SIAM J. Matrix Anal. Appl. 21(4), 1229–1235 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ralph, D.: A stable homotopy approach to horizontal linear complementarity problems. Control Cybern. 31(3), 575–600 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Rapcsák, T.: On the connectedness of the solution set to linear complementarity systems. J. Optim. Theory Appl. 80(3), 501–512 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sznajder, R.: Degree-theoretic analysis of the vertical and horizontal linear complementarity problems. Ph.D. Thesis, University of Maryland Baltimore County (1994)

  20. Sznajder, R., Gowda, M.S.: Generalizations of \(P_0\)- and \(P\)-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223–224, 695–715 (1995)

    Article  MATH  Google Scholar 

  21. Tütüncü, R.H., Todd, M.J.: Reducing horizontal linear complementarity problems. Linear Algebra Appl. 223, 717–729 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Willson, A.N.: A useful generalization of the \(P_0\) matrix concept. Numer. Math. 17, 62–70 (1971)

    Article  MathSciNet  Google Scholar 

  23. Zhang, Y.: On the convergence of a class on infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4(1), 208–227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the editor and the anonymous reviewers for their valuable comments and constructive feedback that have contributed to the improvement of the paper. The first author is a CSIR-SRF fellow, and he wants to thank the Council of Scientific & Industrial Research (CSIR) for the financial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Malaviya National Instiute of Technology Jaipur, Jaipur, 302017, India

    Punit Kumar Yadav & Palpandi Karuppaiah

Authors
  1. Punit Kumar Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Palpandi Karuppaiah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Palpandi Karuppaiah.

Ethics declarations

Conflicts of interest

The authors have no conflict of interests.

Additional information

Communicated by Tibor Illés.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yadav, P.K., Karuppaiah, P. Generalizations of \(R_0\) and \(\textbf{SSM}\) Properties for Extended Horizontal Linear Complementarity Problem. J Optim Theory Appl 199, 392–414 (2023). https://doi.org/10.1007/s10957-023-02262-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02262-9

Keywords

Mathematics Subject Classification

Search

Navigation