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A LOCALLY SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

Published online by Cambridge University Press:  27 March 2015

YU CHEN  and
ZHONG WAN
YU CHEN
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, China email [email protected], [email protected] School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China
ZHONG WAN*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, China email [email protected], [email protected] State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, China
*
✉[email protected]
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Abstract

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We propose a locally smoothing method for some mathematical programs with complementarity constraints, which only incurs a local perturbation on these constraints. For the approximate problem obtained from the smoothing method, we show that the Mangasarian–Fromovitz constraints qualification holds under certain conditions. We also analyse the convergence behaviour of the smoothing method, and present some sufficient conditions such that an accumulation point of a sequence of stationary points for the approximate problems is a C-stationary point, an M-stationary point or a strongly stationary point. Numerical experiments are employed to test the performance of the algorithm developed. The results obtained demonstrate that our algorithm is much more promising than the similar ones in the literature.

Keywords

mathematical programs with complementarity constraintsconstraints qualificationconvergencestationary point

MSC classification

Primary: 90C30: Nonlinear programming
Secondary: 65K10: Optimization and variational techniques
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 3 , January 2015 , pp. 299 - 315
Copyright
© 2015 Australian Mathematical Society 

References

Clarke, F. H., Optimization and Nonsmooth Analysis (Wiley, New York, 1983).Google Scholar
Deng, S.-H., Wan, Z. and Chen, X.-H., “An improved spectral conjugate gradient algorithm for nonconvex unconstrained optimization problems”, J. Optim. Theory Appl. 157 (2013) 820–842; doi:10.1007/s10957-012-0239-7.Google Scholar
Facchinei, F., Jiang, H.-Y. and Qi, L.-Q., “A smoothing method for mathematical programs with equilibrium constraints”, Math. Program. 85 (1999) 107–134; doi:10.1007/s10107990015a.Google Scholar
Fletcher, R., Leyffer, S. and Scholtes, S., “Local convergence of SQP methods for mathematical programs with equilibrium constraints”, SIAM J. Optim. 17 (2006) 259–286; doi:10.1137/S1052623402407382.CrossRefGoogle Scholar
Fukushima, M. and Pang, J.-S., “Convergence of a smoothing continuation method for mathematical programs with complementarity constraints”, in: Ill-posed Variational Problems and Regularization Techniques, 477 Lecture Notes in Economics and Mathematical Systems (eds Théra, M. and Tichatschke, R.), (Springer, Berlin, 1999) 99–110.CrossRefGoogle Scholar
Hoheisel, T., Kanzow, C. and Schwartz, A., “Theoretical and numerical comparison of relaxation scheme for mathematical programs with complementarity constraints”, Math. Program. Ser. A 137 (2013) 257–288; doi:10.1007/s10107-011-0488-5.CrossRefGoogle Scholar
Huang, S., Wan, Z. and Chen, X.-H., “A new nonmonotone line search technique for unconstrained optimization”, Numer. Algor. (2014); doi:10.1007/s11075-014-9866-4.Google Scholar
Huang, S., Wan, Z. and Deng, S. H., “A modified projected conjugate gradient algorithm for unconstrained optimization problems”, ANZIAM J. 54(3) (2013) 143–152; doi:10.1017/S1446181113000084.Google Scholar
Lin, G.-H. and Fukushima, M., “A modified relaxation scheme for mathematical programs with complementarity constraints”, Ann. Oper. Res. 133 (2005) 63–84; doi:10.1007/s10479-004-5024-z.CrossRefGoogle Scholar
Li, Q. and Li, D.-H., “A smoothing Newton method for nonlinear complementarity problems”, Adv. Model. Optim. 13 (2011) 141–152; http://camo.ici.ro/journal/vol13/v13b2.pdf.Google Scholar
Luo, Z.-Q., Pang, J.-S. and Ralph, D., Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, 1996).Google Scholar
Leyffer, S., López-Calva, G. and Nocedal, J., “Interior methods for mathematical programs with complementarity constraints”, SIAM J. Optim. 17 (2006) 52–77; doi:10.1137/040621065.CrossRefGoogle Scholar
Li, Y.-Y., Tan, T. and Li, X.-S., “A log-exponential smoothing method for mathematical programs with complementarity constraints”, Appl. Math. Comput. 218 (2012) 5900–5909; doi:10.1016/j.amc.2011.11.046.Google Scholar
Liu, G.-S., Ye, J.-J. and Zhu, J.-P., “Partial exact penalty for mathematical programs with equilibrium constraints”, Set-Valued Anal. 16 (2008) 785–804; doi:10.1007/s11228-008-0095-7.Google Scholar
Leyffer, S., MacMPEC: AMPL collection of MPECs, http://www.mcs.anl.gov/∼leyffer/MacMPEC/, 2000.Google Scholar
Qi, L.-Q. and Wei, Z.-X., “On the constant positive linear dependence condition and its applications to SQP methods”, SIAM J. Optim. 10 (2000) 963–981; doi:10.1137/S1052623497326629.Google Scholar
Stein, O., “Lifting mathematical programs with complementarity constraints”, Math. Program. Ser. A 131 (2012) 71–94; doi:10.1007/s10107-010-0345-y.Google Scholar
Steffensen, S. and Ulbrich, M., “A new regularization scheme for mathematical programs with equilibrium constraints”, SIAM J. Optim. 20 (2010) 2504–2539; doi:10.1137/090748883.Google Scholar
Wan, Z., Teo, K. L., Shen, X. L. and Hu, C. M., “New BFGS method for unconstrained optimization problem based on modified Armijo line search”, Optimization 63(2) 285–304; doi:10.1080/02331934.2011.644284.Google Scholar
Wan, Z. and Wang, Y.-J., “Convergence of an inexact smoothing method for mathematical programs with equilibrium constraints”, Numer. Funct. Anal. Optim. 27 (2006) 485–495; doi:10.1080/01630560600657323.Google Scholar
Yan, T., “A class of smoothing methods for mathematical programs with complementarity constraints”, Appl. Math. Comput. 186 (2007) 1–9; doi:10.1016/j.amc.2006.05.197.Google Scholar
Yan, T., “A new smoothing scheme for mathematical programs with complementarity constraints”, Sci. China Math. 53 (2010) 1885–1894; doi:10.1007/s11425-010-3080-1.Google Scholar