Notes on some constraint qualifications for mathematical programs with equilibrium constraints.

*(English)*Zbl 1280.90115From the introduction: “Mathematical program with equilibrium constraints (MPEC) plays an important role in many fields such as engineering design, economic equilibria, transportation science, multilevel game, and mathematical programming itself. However, this kind of problems is generally difficult to deal with because their constraints fail to satisfy the standard Mangasarian-Fromovitz constraint qualification at any feasible point.”

In this paper, the authors investigate the weakest constraint qualifications for Bouligand and Mordukhovich stationarities for MPEC and show that there is indeed a gap between Bouligand-stationarity and Mordukhovich-stationarity. They also show that the MPEC relaxed constant positive linear dependence condition can ensure any locally optimal solution to be Mordukhovich stationary. The relations among the existing MPEC constraint qualifications are given, too.

In this paper, the authors investigate the weakest constraint qualifications for Bouligand and Mordukhovich stationarities for MPEC and show that there is indeed a gap between Bouligand-stationarity and Mordukhovich-stationarity. They also show that the MPEC relaxed constant positive linear dependence condition can ensure any locally optimal solution to be Mordukhovich stationary. The relations among the existing MPEC constraint qualifications are given, too.

Reviewer: Nada Djuranović-Miličić (Belgrade)

##### MSC:

90C30 | Nonlinear programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

##### Keywords:

mathematical program with equilibrium constraints; constraint qualification; Bouligand stationarity; Mordukhovich stationarity
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\textit{L. Guo} and \textit{G.-H. Lin}, J. Optim. Theory Appl. 156, No. 3, 600--616 (2013; Zbl 1280.90115)

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##### References:

[1] | Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997) · Zbl 0873.49018 |

[2] | Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) · Zbl 0870.90092 |

[3] | Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers, Boston (1998) · Zbl 0947.90093 |

[4] | Fukushima, M., Lin, G.H.: Smoothing methods for mathematical programs with equilibrium constraints. In: Proceedings of the ICKS’04, pp. 206–213. IEEE Comput. Soc., Los Alamitos (2004) |

[5] | Flegel, M.L., Kanzow, C.: A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, vol. 2, pp. 111–122. Springer, New York (2006) · Zbl 1125.90062 |

[6] | Kanzow, C., Schwartz, A.: Mathematical programs with equilibrium constraints: enhanced fritz john conditions, new constraint qualifications and improved exact penalty results. SIAM J. Optim. 20, 2730–2753 (2010) · Zbl 1208.49016 |

[7] | Scheel, H.S., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000) · Zbl 1073.90557 |

[8] | Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000) · Zbl 1005.49019 |

[9] | Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005) · Zbl 1112.90062 |

[10] | Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997) · Zbl 1088.90042 |

[11] | Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17, 259–286 (2006) · Zbl 1112.90098 |

[12] | Lin, G.H., Guo, L., Ye, J.J.: Solving mathematical programs with equilibrium constraints as constrained equations (submitted) |

[13] | Hu, X., Ralph, D.: Convergence of a penalty method for mathematical programming with equilibrium constraints. J. Optim. Theory Appl. 123, 365–390 (2004) |

[14] | Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001) · Zbl 1010.90086 |

[15] | Guo, L., Lin, G.H., Ye, J.J.: Second order optimality conditions for mathematical programs with equilibrium constraint (submitted) |

[16] | Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006) |

[17] | Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) |

[18] | Flegel, M.L., Kanzow, C.: On M-stationary points for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 310, 286–302 (2005) · Zbl 1136.90484 |

[19] | Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2008) · Zbl 1156.49013 |

[20] | Andreani, R., Haeser, G., Schuverdt, M.L., Siliva, J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. (2011). doi: 10.1007/s10107-011-0456-0 · Zbl 1262.90162 |

[21] | Ye, J.J., Zhang, J.: Enhanced Karush–Kuhn–Tucker condition for mathematical programs with equilibrium constraints (submmited) · Zbl 1320.90086 |

[22] | Flegel, M.L., Kanzow, C.: On the guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54, 517–534 (2005) · Zbl 1147.90397 |

[23] | Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2. Necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009) · Zbl 1205.90294 |

[24] | Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. (2011). doi: 10.1007/s10107-011-0488-5 · Zbl 1262.65065 |

[25] | Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties (submitted) · Zbl 1282.65069 |

[26] | Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21, 314–332 (2011) · Zbl 1229.90216 |

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