# American Institute of Mathematical Sciences

April  2013, 9(2): 305-322. doi: 10.3934/jimo.2013.9.305

## Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 School of Management, Shanghai University, Shanghai 200444, China

Received  January 2012 Revised  May 2012 Published  February 2013

The purpose of the paper is to develop globally convergent algorithms for solving the popular stationarity systems for mathematical programs with complementarity constraints (MPCC) directly. Since the popular stationarity systems for MPCC contain some unknown index sets, we first present some nonsmooth reformulations for the stationarity systems by removing the unknown index sets and then we propose a Levenberg-Marquardt type method to solve them. Under some regularity conditions, we show that the proposed method is globally and superlinearly convergent. We further report some preliminary numerical results.
Citation: Lei Guo, Gui-Hua Lin. Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations. Journal of Industrial & Management Optimization, 2013, 9 (2) : 305-322. doi: 10.3934/jimo.2013.9.305
##### References:
 [1] D. P. Bertsekas, "Nonlinear Programming," $2^{nd}$ edition, Athena Scientific, Belmont, Massachusetts, 1999. Google Scholar [2] F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7 (1997), 225-247. doi: 10.1137/S1052623494279110.  Google Scholar [3] A. Fischer, A special Newton-type optimization method, Optim., 24 (1992), 269-284. doi: 10.1080/02331939208843795.  Google Scholar [4] M. L. Flegel and C. Kanzow, Abadie-type constraint qualification for mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 124 (2005), 595-614. doi: 10.1007/s10957-004-1176-x.  Google Scholar [5] R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM J. Optim., 17 (2006), 259-286. doi: 10.1137/S1052623402407382.  Google Scholar [6] M. Fukushima and G.-H. Lin, Smoothing methods for mathematical programs with equilibrium constraints, Proceedings of the ICKS'04, IEEE Computer Society, (2004), 206-213. Google Scholar [7] L. Guo and G.-H. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints, J. Optim. Theory Appl., (2012). doi: 10.1007/s10957-012-0084-8.  Google Scholar [8] S. Leyffer, MacMPEC-ampl collection of mathematical programms with equilibrium constraints. Available from:, , ().   Google Scholar [9] G.-H. Lin, L. Guo and J.-J. Ye, Solving mathematical programs with equilibrium constraints as constrained equations,, preprint., ().   Google Scholar [10] T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Program., 75 (1996), 407-439. doi: 10.1016/S0025-5610(96)00028-7.  Google Scholar [11] Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar [12] J. V. Outrata, M. Kočvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Theory, Applications and Numerical Results," Nonconvex Optimization and its Applications, 28, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar [13] J.-S. Pang, A B-differentiable equation-baesd, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Program., 51 (1991), 101-131. doi: 10.1007/BF01586928.  Google Scholar [14] J.-S. Pang and A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Program., 60 (1993), 295-337. doi: 10.1007/BF01580617.  Google Scholar [15] J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithm, SIAM J. Optim., 3 (1993), 443-465. doi: 10.1137/0803021.  Google Scholar [16] L. Qi and J. sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275.  Google Scholar [17] P. Tseng, Growth behavior of a class of merit functions for the nonlinear complementarity problem, J. Optim. Theory Appl., 89 (1996), 17-37. doi: 10.1007/BF02192639.  Google Scholar [18] J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369. doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar

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##### References:
 [1] D. P. Bertsekas, "Nonlinear Programming," $2^{nd}$ edition, Athena Scientific, Belmont, Massachusetts, 1999. Google Scholar [2] F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7 (1997), 225-247. doi: 10.1137/S1052623494279110.  Google Scholar [3] A. Fischer, A special Newton-type optimization method, Optim., 24 (1992), 269-284. doi: 10.1080/02331939208843795.  Google Scholar [4] M. L. Flegel and C. Kanzow, Abadie-type constraint qualification for mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 124 (2005), 595-614. doi: 10.1007/s10957-004-1176-x.  Google Scholar [5] R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM J. Optim., 17 (2006), 259-286. doi: 10.1137/S1052623402407382.  Google Scholar [6] M. Fukushima and G.-H. Lin, Smoothing methods for mathematical programs with equilibrium constraints, Proceedings of the ICKS'04, IEEE Computer Society, (2004), 206-213. Google Scholar [7] L. Guo and G.-H. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints, J. Optim. Theory Appl., (2012). doi: 10.1007/s10957-012-0084-8.  Google Scholar [8] S. Leyffer, MacMPEC-ampl collection of mathematical programms with equilibrium constraints. Available from:, , ().   Google Scholar [9] G.-H. Lin, L. Guo and J.-J. Ye, Solving mathematical programs with equilibrium constraints as constrained equations,, preprint., ().   Google Scholar [10] T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Program., 75 (1996), 407-439. doi: 10.1016/S0025-5610(96)00028-7.  Google Scholar [11] Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar [12] J. V. Outrata, M. Kočvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Theory, Applications and Numerical Results," Nonconvex Optimization and its Applications, 28, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar [13] J.-S. Pang, A B-differentiable equation-baesd, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Program., 51 (1991), 101-131. doi: 10.1007/BF01586928.  Google Scholar [14] J.-S. Pang and A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Program., 60 (1993), 295-337. doi: 10.1007/BF01580617.  Google Scholar [15] J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithm, SIAM J. Optim., 3 (1993), 443-465. doi: 10.1137/0803021.  Google Scholar [16] L. Qi and J. sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275.  Google Scholar [17] P. Tseng, Growth behavior of a class of merit functions for the nonlinear complementarity problem, J. Optim. Theory Appl., 89 (1996), 17-37. doi: 10.1007/BF02192639.  Google Scholar [18] J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369. doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar
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