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Equivalence of the generalized complementarity problem to differentiable unconstrained minimization. (English) Zbl 0866.90124
Summary: We consider an unconstrained minimization reformulation of the generalized complementarily 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.
MSC:
 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)
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. [2] Noor, M. A.,Quasi-Complementarity Problem, Journal of Mathematical Analysis and Applications, Vol. 130, pp. 344–353, 1988. · Zbl 0645.90086 · doi:10.1016/0022-247X(88)90310-1 [3] Isac, G.,Complementarity Problems, Lecture Notes in Mathematics 1528, Springer Verlag, Berlin, Germany, 1992. [4] Tseng, P., Yamashita, N., andFukushima, M.,Equivalence of Complementarity Problems to Differentiable Minimization: A Unified Approach, SIAM Journal on Optimization Vol. 6, 1996. [5] Mangasarian, O. L., andSolodov, M. V.,Nonlinear Complementarity as Unconstrained and Constrained Minimization, Mathematical Programming, Vol. 62, pp. 277–297, 1993. · Zbl 0813.90117 · doi:10.1007/BF01585171 [6] Fukushima, M.,Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99–110, 1992. · Zbl 0756.90081 · doi:10.1007/BF01585696 [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. · Zbl 0833.90113 · doi:10.1287/moor.19.4.880 [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. · Zbl 0824.90131 · doi:10.1007/BF02191990 [9] Kanzow, C.,Nonlinear Complementarity as Unconstrained Optimization, Journal of Optimization Theory and Applications, Vol. 88, pp. 139–155, 1996. · Zbl 0845.90120 · doi:10.1007/BF02192026 [10] Jiang, H.,Unconstrained Minimization Approaches to Nonlinear Complementarity Problems, Journal of Global Optimization (to appear). [11] Fischer, A.,A Special Newton-Type Optimization Method, Optimization, Vol. 24, pp. 269–284, 1992. · Zbl 0814.65063 · doi:10.1080/02331939208843795 [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. [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. · Zbl 0247.65038 · doi:10.1016/0024-3795(73)90006-2 [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. · Zbl 0734.90098 · doi:10.1007/BF01582255 [15] Cottle, R. W., Pang, J. S., andStone, R. E.,The Linear Complementarity Problem, Academic Press, Boston, Massachusetts, 1992. [16] Facchinei, F., andSoares, J.,A New Merit Function for Nonlinear Complementarity Problems and a Related Algorithm, SIAM Journal on Optimization (to appear). [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. · Zbl 0827.90131 · doi:10.1137/S0363012992241673 [18] Jiang, H., andQi, L.,A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems, SIAM Journal on Control and Optimization (to appear). [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. [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. · doi:10.1287/moor.12.3.474 [21] Mangasarian, O. L., andRen, J.,New Improved Error Bounds for the Linear Complementarity Problem, Mathematical Programming, Vol. 66, pp. 241–255, 1994. · Zbl 0829.90124 · doi:10.1007/BF01581148 [22] Luo, X. D., andTseng, P.,On Global Projection-Type Error Bound for the Linear Complementarity Problem, Linear Algebra and Its Applications (to appear). [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. · Zbl 0866.90127 · doi:10.1007/BF02192639 [24] Pang, J. S., Private Communication, 1995. [25] Pang, J. S.,Inexact Newton Methods for the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 36, pp. 54–71, 1986. · Zbl 0613.90097 · doi:10.1007/BF02591989