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Journal of Global Optimization

, Volume 73, Issue 1, pp 153–169 | Cite as

The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

  • Xiaoni Chi
  • M. Seetharama GowdaEmail author
  • Jiyuan Tao
Article
  • 46 Downloads

Abstract

A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of \(\mathbf{R}_0\), \(\mathbf{R}\), and \(\mathbf{P}\) properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of \({\mathbb {R}}^{n}\). We show how our results naturally lead to interior point systems.

Keywords

Weighted horizontal linear complementarity problem Euclidean Jordan algebra Degree \(\mathbf{R}_0\)-pair 

Mathematics Subject Classification

90C30 

Notes

Acknowledgements

The work of the first author is supported by the National Natural Science Foundation of China (No. 11401126) and Guangxi Natural Science Foundation (Nos. 2016GXNSFBA380102, 2014GXNSFFA118001), China. The third author was supported by Loyola Summer Research Grant 2017.

References

  1. 1.
    Anstreicher, K.M.: Interior-point algorithms for a generalization of linear programming and weighted centering. Optim. Methods Softw. 27(4–5), 605–612 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cottle, R.W., Pang, J.-S., Stone, R.: The Linear Complementarity Problem. Academic Press, Boston (1992)zbMATHGoogle Scholar
  3. 3.
    Eisenberg, E., Gale, D.: Consensus of subjective probabilities: the pari-mutuel method. Ann. Math. Statist. 30, 165–168 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)zbMATHGoogle Scholar
  5. 5.
    Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)zbMATHGoogle Scholar
  6. 6.
    Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math. Oper. Res. 18, 868–879 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kojima, M., Mizuno, S., Noma, T.: Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems. Math. Oper. Res. 15(4), 662–675 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kojima, M., Megiddo., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science 538, Springer-Verlag, Berlin (1991)Google Scholar
  10. 10.
    Lloyd, N.G.: Degree Theory. Cambridge University Press, London (1978)zbMATHGoogle Scholar
  11. 11.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York (1970)zbMATHGoogle Scholar
  12. 12.
    Ouellette, D.V.: Schur complements and statistics. Linear Algebra Appl. 36, 187–295 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Potra, F.A.: Weighted complementarity problems—a new paradigm for computing equilibria. SIAM J. Optim. 22(4), 1634–1654 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Potra, F.A.: Sufficient weighted complementarity problems. Comput. Optim. Appl. 64, 467–488 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sznajder, R.: Degree-theoretic analysis of the vertical and horizontal linear complementarity problems, Ph.D. Thesis, University of Maryland Baltimore County (1994)Google Scholar
  16. 16.
    Sznajder, R., Gowda, M.S.: Generalizations of \(P_0\),(P)-properties; extended vertical and horizontal LCPs. Linear Algebra Appl. 223(224), 695–715 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ye, Y.: Math. Program. A path to the Arrow–Debreu competitive market equilibrium. 111(1–2), 315–348 (2008)Google Scholar
  18. 18.
    Yoshise, A.: Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM J. Optim. 17, 1129–1153 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and ComputationGuilin University of Electronic TechnologyGuilinPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA
  3. 3.Department of Mathematics and StatisticsLoyola University MarylandBaltimoreUSA

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