Exact penalization and necessary optimality conditions for multiobjective optimization problems with equilibrium constraints. (English) Zbl 1474.90433

Summary: A calmness condition for a general multiobjective optimization problem with equilibrium constraints is proposed. Some exact penalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to the calmness condition. Subsequently, a Mordukhovich stationary necessary optimality condition based on the exact penalization results is obtained. Moreover, some applications to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints are given.


90C29 Multi-objective and goal programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C46 Optimality conditions and duality in mathematical programming
Full Text: DOI


[1] Luo, Z.; Pang, J.; Ralph, D.; Wu, S., Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints, Mathematical Programming B, 75, 1, 19-76 (1996) · Zbl 0870.90092
[2] Ye, J. J.; Ye, X. Y., Necessary optimality conditions for optimization problems with variational inequality constraints, Mathematics of Operations Research, 22, 4, 977-997 (1997) · Zbl 1088.90042
[3] Ye, J. J.; Zhu, Q. J., Multiobjective optimization problem with variational inequality constraints, Mathematical Programming B, 96, 1, 139-160 (2003) · Zbl 1041.90052
[4] Flegel, M. L.; Kanzow, C., On M-stationary points for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 310, 1, 286-302 (2005) · Zbl 1136.90484
[5] Chen, Y.; Florian, M., The nonlinear bilevel programming problem: formulations, regularity and optimality conditions, Optimization, 32, 120-145 (1995) · Zbl 0817.90101
[6] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), New York, NY, USA: Wiley, New York, NY, USA · Zbl 0582.49001
[7] Rockafellar, R. T.; Wets, R. J. B., Variational Analysis (1998), Berlin, Germany: Springer, Berlin, Germany · Zbl 0888.49001
[8] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation, Volume I: Basic Theory, Volume II: Applications (2006), Berlin, Germany: Springer, Berlin, Germany
[9] Burke, J. V., Calmness and exact penalization, SIAM Journal on Control and Optimization, 29, 2, 493-497 (1991) · Zbl 0734.90090
[10] Fiacco, A.; McCormic, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques (1987), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[11] Rubinov, A. M.; Glover, B. M.; Yang, X. Q., Decreasing functions with applications to penalization, SIAM Journal on Optimization, 10, 1, 289-313 (1999) · Zbl 0955.90127
[12] Rubinov, A. M.; Yang, X. Q., Lagrange-Type Functions in Constrained Non-Convex Optimzation (2003), New York, NY, USA: Kluwer Academic Publishers, New York, NY, USA
[13] Huang, X. X.; Yang, X. Q., Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization, SIAM Journal on Optimization, 13, 3, 675-692 (2002) · Zbl 1036.90062
[14] Bao, T. Q.; Gupta, P.; Mordukhovich, B. S., Necessary conditions in multiobjective optimization with equilibrium constraints, Journal of Optimization Theory and Applications, 135, 2, 179-203 (2007) · Zbl 1146.90508
[15] Robinson, S. M., Stability theory for systems of inequalities, part II: differentiable nonlinear systems, SIAM Journal on Numerical Analysis, 13, 4, 497-513 (1976) · Zbl 0347.90050
[16] Treiman, J. S., The linear nonconvex generalized gradient and Lagrange multipliers, SIAM Journal on Optimization, 5, 670-680 (1995) · Zbl 0829.49017
[17] Gopfert, A.; Riahi, H.; Tammer, C.; Zalinescu, C., Variational Methods in Partially Ordered Spaces (2003), Berlin, Germany: Springer, Berlin, Germany · Zbl 1140.90007
[18] Chen, G. Y.; Yang, X. Q., Characterizations of variable domination structures via nonlinear scalarization, Journal of Optimization Theory and Applications, 112, 1, 97-110 (2002) · Zbl 0988.49005
[19] Durea, M.; Tammer, C., Fuzzy necessary optimality conditions for vector optimization problems, Optimization, 58, 4, 449-467 (2009) · Zbl 1162.90597
[20] Chen, C. R.; Li, S. J.; Fang, Z. M., On the solution semicontinuity to a parametric generalized vector quasivariational inequality, Computers and Mathematics with Applications, 60, 8, 2417-2425 (2010) · Zbl 1205.49036
[21] Dontchev, A. L.; Rockafellar, R. T., Implicit Functions and Solution Mappings (2009), Dordrecht, The Netherlands: Springer, Dordrecht, The Netherlands · Zbl 1172.49013
[22] Ioffe, A. D., Metric regularity and subdifferential calculus, Russian Mathematical Surveys, 55, 3, 501-558 (2000) · Zbl 0979.49017
[23] Henrion, R.; Jourani, A.; Outrata, J., On the calmness of a class of multifunction, SIAM Journal on Optimization, 13, 2, 603-618 (2002) · Zbl 1028.49018
[24] Zheng, X. Y.; Ng, K. F., Metric subregularity and constraint qualifications for convex generalized equations in banach spaces, SIAM Journal on Optimization, 18, 2, 437-460 (2007) · Zbl 1190.90230
[25] Jahn, J., Vecor Optimization, Theory, Applications and Extension (2004), Berlin, Germany: Springer, Berlin, Germany
[26] Kanzow, C.; Schwartz, A., Mathematical programs with equilibrium constraints: enhanced Fritz John-conditions, new constraint qualifications, and improved exact penalty results, SIAM Journal on Optimization, 20, 5, 2730-2753 (2010) · Zbl 1208.49016
[27] Ye, J. J.; Zhang, J., Enhanced Karush-Kuhn-Tucker conditions for mathematical programs with equilibrium constraints, Journal of Optimization Theory and Applications (2013) · Zbl 1320.90086
[28] Schwartz, A., Mathematical programs with complementarity constraints: theory, methods, and applications [Ph.D. dissertation] (2011), Institute of Applied Mathematics and Statistics, University of Wurzburg
[29] Guo, L.; Ye, J. J.; Zhang, J., Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity, SIAM Journal on Optimization, 23, 2295-2319 (2013) · Zbl 1342.90144
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.