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Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems

Author

Listed:
  • Lingchen Kong

    (Beijing Jiaotong University)

  • Levent Tunçel

    (University of Waterloo)

  • Naihua Xiu

    (Beijing Jiaotong University)

Abstract

We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem. Employing an algebraic characterization of homogeneous cones due to Vinberg from the 1960s, we generalize the properties of existence and uniqueness of solutions for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of homogeneous cone complementarity problem. We provide sufficient conditions for a continuous function so that the associated homogeneous cone complementarity problems have solutions. In particular, we give sufficient conditions for a monotone continuous function so that the associated homogeneous cone complementarity problem has a unique solution (if any). Moreover, we establish a global error bound for the homogeneous cone complementarity problem under some conditions.

Suggested Citation

  • Lingchen Kong & Levent Tunçel & Naihua Xiu, 2012. "Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 357-376, May.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:2:d:10.1007_s10957-011-9971-7
    DOI: 10.1007/s10957-011-9971-7
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    References listed on IDEAS

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    1. Levent Tunçel & Song Xu, 2001. "On Homogeneous Convex Cones, The Carathéodory Number, and the Duality Mapping," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 234-247, May.
    2. M. Seetharama Gowda & Roman Sznajder, 2006. "Automorphism Invariance of P - and GUS -Properties of Linear Transformations on Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 109-123, February.
    3. M. Seetharama Gowda, 1993. "Applications of Degree Theory to Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 868-879, November.
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