# Equivalence of the generalized complementarity problem to differentiable unconstrained minimization

## Abstract

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 problem nonlinear complementarity problem unconstrained minimization stationary point bounded level set global error bound## Preview

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## References

- 1.Pang, J. S.,
*The Implicit Complementarity Problem*4, Nonlinear Programming 4, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 487–518, 1981.Google Scholar - 2.Noor, M. A.,
*Quasi-Complementarity Problem*, Journal of Mathematical Analysis and Applications, Vol. 130, pp. 344–353, 1988.CrossRefGoogle Scholar - 3.Isac, G.,
*Complementarity Problems*, Lecture Notes in Mathematics 1528, Springer Verlag, Berlin, Germany, 1992.Google Scholar - 4.Tseng, P., Yamashita, N., andFukushima, M.,
*Equivalence of Complementarity Problems to Differentiable Minimization: A Unified Approach*, SIAM Journal on Optimization Vol. 6, 1996.Google Scholar - 5.Mangasarian, O. L., andSolodov, M. V.,
*Nonlinear Complementarity as Unconstrained and Constrained Minimization*, Mathematical Programming, Vol. 62, pp. 277–297, 1993.CrossRefGoogle Scholar - 6.Fukushima, M.,
*Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems*, Mathematical Programming, Vol. 53, pp. 99–110, 1992.CrossRefGoogle Scholar - 7.Luo, Z. Q., Mangasarian, O. L., Ren, J., andSolodov, M. V.,
*New Error Bounds for the Linear Complementarity Problem*, Mathematics of Operations Research, Vol. 19, pp. 880–892, 1994.Google Scholar - 8.Yamashita, N., andFukushima, M.,
*On Stationary Points of the Implicit Lagrangian for Nonlinear Complementarity Problems*, Journal of Optimization Theory and Applications, Vol. 84, pp. 653–663, 1995.CrossRefGoogle Scholar - 9.Kanzow, C.,
*Nonlinear Complementarity as Unconstrained Optimization*, Journal of Optimization Theory and Applications, Vol. 88, pp. 139–155, 1996.Google Scholar - 10.Jiang, H.,
*Unconstrained Minimization Approaches to Nonlinear Complementarity Problems*, Journal of Global Optimization (to appear).Google Scholar - 11.Fischer, A.,
*A Special Newton-Type Optimization Method*, Optimization, Vol. 24, pp. 269–284, 1992.Google Scholar - 12.Fischer, A.,
*An NCP-Function and Its Use for the Solution of Complementarity Problems*, Recent Advances in Nonsmooth Optimization, Edited by D. Z. Du, L. Qi, and R. S. Womersley, World Scientific Publishers, Singapore, Republic of Singapore, pp. 88–105, 1995.Google Scholar - 13.Moré, J. J., andRheinboldt, W. C.,
*On P and S-Functions and Related Classes of n-Dimensional Nonlinear Mappings*, Linear Algebra and Its Applications, Vol. 6, pp. 45–68, 1973.CrossRefGoogle Scholar - 14.Harker, P. T., andPang, J. S.,
*Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications*, Mathematical Programming, Vol. 48, pp. 161–220, 1990.CrossRefGoogle Scholar - 15.Cottle, R. W., Pang, J. S., andStone, R. E.,
*The Linear Complementarity Problem*, Academic Press, Boston, Massachusetts, 1992.Google Scholar - 16.Facchinei, F., andSoares, J.,
*A New Merit Function for Nonlinear Complementarity Problems and a Related Algorithm*, SIAM Journal on Optimization (to appear).Google Scholar - 17.Pang, J. S., andYao, J. C.,
*On a Generalization of a Normal Map and Equation*, SIAM Journal on Control and Optimization, Vol. 33, pp. 168–184, 1995.CrossRefGoogle Scholar - 18.Jiang, H., andQi, L.,
*A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems*, SIAM Journal on Control and Optimization (to appear).Google Scholar - 19.Mangasarian, O. L., andRen, J.,
*New Error Bounds for the Nonlinear Complementarity Problem*, Communications on Applied Nonlinear Analysis, Vol. 1, pp. 49–56, 1994.Google Scholar - 20.Pang, J. S.,
*A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem*, Mathematics of Operations Research, Vol. 12, pp. 474–484, 1987.Google Scholar - 21.Mangasarian, O. L., andRen, J.,
*New Improved Error Bounds for the Linear Complementarity Problem*, Mathematical Programming, Vol. 66, pp. 241–255, 1994.CrossRefGoogle Scholar - 22.Luo, X. D., andTseng, P.,
*On Global Projection-Type Error Bound for the Linear Complementarity Problem*, Linear Algebra and Its Applications (to appear).Google Scholar - 23.Tseng, P.,
*Growth Behaviour of a Class of Merit Functions for the Nonlinear Complementarity Problem*, Journal of Optimization Theory and Applications, Vol. 89, pp. 17–37, 1996.CrossRefGoogle Scholar - 24.Pang, J. S., Private Communication, 1995.Google Scholar
- 25.Pang, J. S.,
*Inexact Newton Methods for the Nonlinear Complementarity Problem*, Mathematical Programming, Vol. 36, pp. 54–71, 1986.Google Scholar