Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints.

*(English)*Zbl 1412.90069Summary: This paper focuses on solving normalized stationary points of a class of equilibrium problem with equilibrium constraints (EPEC). We show that, under some kind of separability assumption, normalized C-/M-/S-stationary points of EPEC are actually C-/M-/S-stationary points of an associated mathematical program with equilibrium constraints (MPEC), which implies that we can solve MPEC to obtain normalized stationary points of EPEC. In addition, we demonstrate the proposed approach on competition of manufacturers for similar products in the same city.

##### MSC:

90B50 | Management decision making, including multiple objectives |

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

65K10 | Numerical optimization and variational techniques |

##### Keywords:

equilibrium problem with equilibrium constraints; mathematical program with equilibrium constraints; stationary point##### Software:

MacMPEC
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\textit{P. Li}, J. Ind. Manag. Optim. 14, No. 2, 637--646 (2018; Zbl 1412.90069)

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##### References:

[1] | F. Facchinei; A. Fischer; V. Piccialli, On generalized Nash games and variational inequalities, Operations Research Letters, 35, 159, (2007) · Zbl 1303.91020 |

[2] | J. Y. Fan; Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74, 23, (2005) · Zbl 1076.65047 |

[3] | M. L. Flegel; C. Kanzow, On M-stationary points for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 310, 286, (2005) · Zbl 1136.90484 |

[4] | R. Fletcher; S. Leyffer, Solving mathematical programs with complementarity constraints as nonlinear programs, Optimization Methods and Software, 19, 15, (2004) · Zbl 1074.90044 |

[5] | R. Fletcher; S. Leyffer; D. Ralph; S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 17, 259, (2006) · Zbl 1112.90098 |

[6] | L. Guo; G. H. Lin, Global algorithm for solving stationary points for equilibrium programs with shared equilibrium constraints, Pacific Journal of Optimization, 9, 443, (2013) · Zbl 1277.90131 |

[7] | L. Guo; G. H. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints, Journal of Optimization Theory and Applications, 156, 600, (2013) · Zbl 1280.90115 |

[8] | L. Guo; G. H. Lin; D. Zhang; D. Zhu, An MPEC reformulation of an EPEC model for electricity markets, Operations Research Letters, 43, 262, (2015) · Zbl 1408.91164 |

[9] | P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operations Research, 54, 81, (1991) · Zbl 0754.90070 |

[10] | M. Hu; M. Fukushima, Variational inequality formulation of a class of multi-leader-follower games, Journal of Optimization Theory and Applications, 151, 455, (2011) · Zbl 1231.91049 |

[11] | M. Hu; M. Fukushima, Existence, uniqueness, and computation of robust Nash equilibrium in a class of multi-leader-follower games, SIAM Journal on Optimization, 23, 894, (2013) · Zbl 1273.91094 |

[12] | X. Hu, Mathematical Programs with Complementarity Constraints and Game Theory Models in Electricity Markets, Ph. D thesis, Department of Mathematics and Statistics, University of Melbourne, 2003. |

[13] | A. A. Kulkarni; U. V. Shanbhag, A shared-constraint approach to multi-leader multi-follower games, Set-Valued and Variational Analysis, 22, 691, (2014) · Zbl 1307.91046 |

[14] | A. A. Kulkarni; U. V. Shanbhag, On the consistency of leaders’ conjectures in hierarchical games, 52nd IEEE Annual Conference on Decision and Control (CDC), , 1180, (2013) |

[15] | S. Leyffer; T. Munson, Solving multi-leader-common-follower games, Optimization Methods and Software, 25, 601, (2010) · Zbl 1201.90187 |

[16] | Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996. · Zbl 0898.90006 |

[17] | B. S. Mordukhovich, Optimization and equilibrium problems with equilibrium constraints in infinite-dimensional spaces, Optimization, 57, 715, (2008) · Zbl 1161.49015 |

[18] | J. V. Outrata, A note on a class of equilibrium problems with equilibrium constraints, Kybernetika, 40, 585, (2004) · Zbl 1249.49017 |

[19] | J. S. Pang; M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Computational Management Science, 2, 21, (2005) · Zbl 1115.90059 |

[20] | J. B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33, 520, (1965) · Zbl 0142.17603 |

[21] | C. L. Su, A sequential NCP algorithm for solving equilibrium problems with equilibrium constraints, Technical Report, Department of Management Science and Engineering, Stanford University, 2004. |

[22] | H. Scheel; S. Scholtes, Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity, Mathematics of Operations Research, 25, 1, (2000) · Zbl 1073.90557 |

[23] | S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11, 918, (2001) · Zbl 1010.90086 |

[24] | J. J. Ye, Optimality conditions for optimization problems with complementarity constraints, SIAM Journal on Optimization, 9, 374, (1999) · Zbl 0967.90092 |

[25] | J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 307, 350, (2005) · Zbl 1112.90062 |

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