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A new smoothing method for mathematical programs with complementarity constraints based on logarithm-exponential function. (English) Zbl 1427.90259
Summary: We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

##### MSC:
 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
MacMPEC
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##### References:
 [1] Luo, Z.; Pang, J.; Ralph, D., Mathematical Programs with Equilibrium Constraints, (1996), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0898.90006 [2] Wan, Z., Further investigation on feasibility of mathematical programs with equilibrium constraints, Computers & Mathematics with Applications. An International Journal, 44, 1-2, 7-11, (2002) · Zbl 1008.90063 [3] Fletcher, R.; Leyffer, S.; Ralph, D.; Scholtes, S., Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 17, 1, 259-286, (2006) · Zbl 1112.90098 [4] Fukushima, M.; Luo, Z.-Q.; Pang, J.-S., A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Computational optimization and applications, 10, 1, 5-34, (1998) · Zbl 0904.90153 [5] Jiang, H.; Ralph, D., Smooth SQP methods for mathematical programs with nonlinear complementarity constraints, SIAM Journal on Optimization, 10, 3, 779-808, (2000) · Zbl 0955.90134 [6] Jian, J.-B., A superlinearly convergent implicit smooth SQP algorithm for mathematical programs with nonlinear complementarity constraints, Computational Optimization and Applications, 31, 3, 335-361, (2005) · Zbl 1122.90095 [7] Li, J.-L.; Jian, J.-B., A superlinearly convergent SSLE algorithm for optimization problems with linear complementarity constraints, Journal of Global Optimization, 33, 4, 477-510, (2005) · Zbl 1097.90063 [8] Jian, J.-B.; Li, J.-L.; Mo, X.-D., A strongly and superlinearly convergent SQP algorithm for optimization problems with linear complementarity constraints, Applied Mathematics & Optimization, 54, 1, 17-46, (2006) · Zbl 1136.90514 [9] Leyffer, S.; López-Calva, G.; Nocedal, J., Interior methods for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 17, 1, 52-77, (2006) · Zbl 1112.90095 [10] Raghunathan, A. U.; Biegler, L. T., An interior point method for mathematical programs with complementarity constraints (MPCCs), SIAM Journal on Optimization, 15, 3, 720-750, (2005) · Zbl 1077.90079 [11] Lin, G. H.; Fukushima, M., Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, Journal of Optimization Theory and Applications, 118, 1, 67-80, (2003) · Zbl 1033.90087 [12] Liu, G.; Ye, J.; Zhu, J., Partial exact penalty for mathematical programs with equilibrium constraints, Set-Valued Analysis, 16, 5-6, 785-804, (2008) · Zbl 1170.65052 [13] Jiang, S.; Zhang, J.; Chen, C.; Lin, G., Smoothing partial exact penalty splitting method for mathematical programs with equilibrium constraints, Journal of Global Optimization, 70, 1, 223-236, (2018) · Zbl 1394.90469 [14] Stein, O., Lifting mathematical programs with complementarity constraints, Mathematical Programming, 131, 1-2, Ser. A, 71-94, (2012) · Zbl 1250.90094 [15] Scholtes, S., Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11, 4, 918-936, (2001) · Zbl 1010.90086 [16] Lin, G.-H.; Fukushima, M., A modified relaxation scheme for mathematical programs with complementarity constraints, Annals of Operations Research, 133, 63-84, (2005) · Zbl 1119.90058 [17] Kadrani, A.; Dussault, J.-P.; Benchakroun, A., A new regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 20, 1, 78-103, (2009) · Zbl 1187.65064 [18] Steffensen, S.; Ulbrich, M., A new relaxation scheme for mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 20, 5, 2504-2539, (2010) · Zbl 1231.90350 [19] Hoheisel, T.; Kanzow, C.; Schwartz, A., Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints, Mathematical Programming, 137, 1-2, Ser. A, 257-288, (2013) · Zbl 1262.65065 [20] Dussault, J. P.; Haddou, M.; Migot, T., The new butterfly relaxation methods for mathematical programs with complementarity constraints · Zbl 1421.90137 [21] Ramos, A., Mathematical programs with equilibrium constraints: A sequential optimality condition, new constraint qualifications and algorithmic consequences, Technical report, (2016) [22] Migot, T.; Dussault, J. P.; Haddou, M.; Kadrani, A., How to compute a local minimum of the MPCC [23] Facchinei, F.; Jiang, H.; Qi, L., A smoothing method for mathematical programs with equilibrium constraints, Mathematical Programming, 85, 1, Ser. A, 107-134, (1999) · Zbl 0959.65079 [24] Zhu, Z.-b.; Luo, Z.-j.; Zeng, J.-w., A new smoothing technique for mathematical programs with equilibrium constraints, Applied Mathematics and Mechanics-English Edition, 28, 10, 1407-1414, (2007) · Zbl 1231.90353 [25] Yin, H.; Zhang, J., Global convergence of a smooth approximation method for mathematical programs with complementarity constraints, Mathematical Methods of Operations Research, 64, 2, 255-269, (2006) · Zbl 1132.90370 [26] Wan, Z.; Wang, Y., Convergence of an inexact smoothing method for mathematical programs with equilibrium constraints, Numerical Functional Analysis and Optimization, 27, 3-4, 485-495, (2006) · Zbl 1136.90479 [27] Li, Y.; Tan, T.; Li, X., A log-exponential smoothing method for mathematical programs with complementarity constraints, Applied Mathematics and Computation, 218, 10, 5900-5909, (2012) · Zbl 1248.65065 [28] Yan, T., A class of smoothing methods for mathematical programs with complementarity constraints, Applied Mathematics and Computation, 186, 1, 1-9, (2007) · Zbl 1116.65074 [29] Yan, T., A new smoothing scheme for mathematical programs with complementarity constraints, Science China Mathematics, 53, 7, 1885-1894, (2010) · Zbl 1209.90323 [30] Chen, Y.; Wan, Z., A smoothing regularization method for mathematical programs with complementarity constraints with strong convergent property, Pacific Journal of Optimization, 12, 3, 497-519, (2016) · Zbl 1352.65150 [31] Wan, Z.; Yuan, M.; Wang, C., A partially smoothing Jacobian method for nonlinear complementarity problems with P_0 function, Journal of Computational and Applied Mathematics, 286, 158-171, (2015) · Zbl 1315.65053 [32] Li, D. H.; Tong, X. J.; Wan, Z., Numerical Optomization, (2005), Beijing, China: Science Press, Beijing, China [33] Mangasarian, O. L., Nonlinear programming, (1969), NY, USA: McGraw-Hill, NY, USA · Zbl 0194.20201 [34] Andreani, R.; Martinez, Y. M.; Ramos, A.; Silva, P. J., A cone-continuity constraint qualifications and algorithmic consequences, SIAM Journal on Optimization, 26, 1, 96-110, (2016) · Zbl 1329.90162 [35] Rockafellar, R. T.; Wets, R. J., Variational analysis, volume 317, (2009), Springer Science Business Media [36] Bertsekas, D. P., Minimax methods based on approximation, Proceedings of the 1976 Johns Hopkins Conference on information Sciences and Systems [37] Deng, S.; Wan, Z., An improved three-term conjugate gradient algorithm for solving unconstrained optimization problems, Optimization. A Journal of Mathematical Programming and Operations Research, 64, 12, 2679-2691, (2015) · Zbl 1327.90303 [38] Huang, S.; Wan, Z.; Chen, X., A new nonmonotone line search technique for unconstrained optimization, Numerical Algorithms, 68, 4, 671-689, (2015) · Zbl 1309.90100 [39] Wan, Z.; Teo, K. L.; Shen, X.; Hu, C., New BFGS method for unconstrained optimization problem based on modified Armijo line search, Optimization. A Journal of Mathematical Programming and Operations Research, 63, 2, 285-304, (2014) · Zbl 1288.90096 [40] Jian, J.; Han, L.; Jiang, X., A hybrid conjugate gradient method with descent property for unconstrained optimization, Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, 39, 3-4, 1281-1290, (2015) [41] Qi, L.; Wei, Z., On the constant positive linear dependence condition and its application to SQP methods, SIAM Journal on Optimization, 10, 4, 963-981, (2000) · Zbl 0999.90037 [42] Leyffer, S., MacMPEC, AMPL collection of MPECs
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