Abstract
In this paper, we consider a mathematical program with complementarity constraints. We present a modified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification holds for the new relaxed problem under some mild conditions. We also consider a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is C-stationary to the original problem under the MPEC linear independence constraint qualification and, if the Hessian matrices of the Lagrangian functions of the relaxed problems are uniformly bounded below on the corresponding tangent space, it is M-stationary. We also obtain some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices mentioned above are new and can be verified easily.
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References
Chen, X. and M. Fukushima. (2004). “A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints.” Computational Optimization and Applications 27, 223–246.
Chen, Y. and M. Florian. (1995). “The Nonlinear Bilevel Programming Problem: Formulations, Regularity and Optimality Conditions.” Optimization 32, 193–209.
Cottle, R.W., J.S. Pang, and R.E. Stone. (1992). The Linear Complementarity Problem. New York, NY: Academic Press.
Facchinei, F., H. Jiang, and L. Qi. (1999). “A Smoothing Method for Mathematical Programs with Equilibrium Constraints.” Mathematical Programming 85, 107–134.
Fukushima, M., Z.Q. Luo, and J.S. Pang. (1998). “A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints.” Computational Optimization and Applications 10, 5–34.
Fukushima, M. and J.S. Pang. (1999). “Convergence of a Smoothing Continuation Method for Mathematical Problems with Complementarity Constraints.” In M. Théra and R. Tichatschke (eds.), Illposed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems, Vol. 477. Berlin/Heidelberg: Springer, pp. 105–116.
Fukushima, M. and J.S. Pang. (1998). “Some Feasibility Issues in Mathematical Programs with Equilibrium Constraints.” SIAM Journal on Optimization 8, 673–681.
Hu, X. and D. Ralph. (2004). “Convergence of a Penalty Method for Mathematical Programming with Equilibrium Constraints.” Journal of Optimization Theory and Applications 123, 365–390.
Huang, X.X., X.Q. Yang, and D.L. Zhu. (2001). “A Sequential Smooth Penalization Approach to Mathematical Programs with Complementarity Constraints.” Manuscript, Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong.
Jiang, H. and D. Ralph. (2000). “Smooth Methods for Mathematical Programs with Nonlinear Complementarity Constraints.” SIAM Journal on Optimization 10, 779–808.
Lin, G.H. and M. Fukushima. (2003). “New Relaxation Method for Mathematical Programs with Complementarity Constraints.” Journal of Optimization Theory and Applications 118, 81–116.
Luo, Z.Q., J.S. Pang, and D. Ralph. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge: Cambridge University Press.
Luo, Z.Q., J.S. Pang, D. Ralph, and S.Q. Wu. (1996). “Exact Penalization and Stationary Conditions of Mathematical Programs with Equilibrium Constraints.” Mathematical Programming 75, 19–76.
Ortega, J.M. and W.C. Rheinboldt. (1970). Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press.
Outrata, J.V. (1994). “On Optimization Problems with Variational Inequality Constraints.” SIAM Journal on Optimization 4, 340–357.
Outrata, J.V. and J. Zowe. (1995). “A Numerical Approach to Optimization Problems with Variational Inequality Constraints.” Mathematical Programming 68, 105–130.
Pang, J.S. and M. Fukushima. (1999). “Complementarity Constraint Qualifications and Simplified B-Stationarity Conditions for Mathematical Programs with Equilibrium Constraints.” Computational Optimization and Applications 13, 111–136.
Scheel, H. and S. Scholtes. (2000). “Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensivity.” Mathematics of Operations Research 25, 1–22.
Scholtes, S. (2001). “Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints.” SIAM Journal on Optimization 11, 918–936.
Scholtes, S. and M. Stöhr. (1999). “Exact Penalization of Mathematical Programs with Complementarity Constraints.” SIAM Journal on Control and Optimization 37, 617–652.
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This work was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan. The authors are grateful to an anonymous referee for critical comments.
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Lin, GH., Fukushima, M. A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints. Ann Oper Res 133, 63–84 (2005). https://doi.org/10.1007/s10479-004-5024-z
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DOI: https://doi.org/10.1007/s10479-004-5024-z