Journal of Optimization Theory and Applications

, Volume 90, Issue 3, pp 581–603

Equivalence of the generalized complementarity problem to differentiable unconstrained minimization

  • C. Kanzow
  • M. Fukushima
Contributed Papers

DOI: 10.1007/BF02189797

Cite this article as:
Kanzow, C. & Fukushima, M. J Optim Theory Appl (1996) 90: 581. doi:10.1007/BF02189797


We consider an unconstrained minimization reformulation of the generalized complementarity problem (GCP). The merit function introduced here is differentiable and has the property that its global minimizers coincide with the solutions of GCP. Conditions for its stationary points to be global minimizers are given. Moreover, it is shown that the level sets of the merit function are bounded under suitable assumptions. We also show that the merit function provides global error bounds for GCP. These results are based on a condition which reduces to the condition of the uniform P-function when GCP is specialized to the nonlinear complementarity problem. This condition also turns out to be useful in proving the existence and uniqueness of a solution for GCP itself. Finally, we obtain as a byproduct an error bound result with the natural residual for GCP.

Key Words

Generalized complementarity problemnonlinear complementarity problemunconstrained minimizationstationary pointbounded level setglobal error bound

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • C. Kanzow
    • 1
  • M. Fukushima
    • 2
  1. 1.Institute of Applied MathematicsUniversity of HamburgHamburgGermany
  2. 2.Department of Applied Mathematics and Physics, Graduate School of EngineeringKyoto UniversityKyotoJapan