均衡约束数学规划的约束规格和最优性条件综述

Abstract

Abstract: This is a survey on constraint qualifications and optimality conditions for mathematical programs with equilibrium constraints (MPEC for short). Some important international research results on constraint qualifications and the corresponding optimality conditions for MPEC are introduced. The context included are as follows: (1) Some constraint qualifications in common use for MPEC (e.g., MPEC-LICQ, MPEC-MFCQ) and some latest developed constraint qualifications (e.g., constant rank constraint qualifications), and their relationships; (2) Various stationary points for MPEC; (3) Optimality conditions for MPEC. Finally, we discuss some future research perspectives of constraint qualifications and optimality conditions for MPEC.

Key words: equilibrium constraint, mathematical programs, constraint qualification, stationary points, optimality conditions

CLC Number:

O221

LI Jianling, XIE Qin, JIAN Jinbao. Review on constraint qualifications and optimality conditions for mathematical programs with equilibrium constraints[J]. Operations Research Transactions, 2013, 17(3): 73-85.

References

Luo Z Q, Pang J S, Ralph D. Mathematical Programs with Equilibrium Constraints [M]. Cambridge: Cambridge University Press. 1996.
Outrata J V, Kovara M, Zowe J. Nonsmooth approach to optimization problems with equilibrium constraints [M]//Noneonvex Optimization and its Applications, Dordreeht: Kluwer Aeademie Publishers, 1998.
Flegel M L, Kanzow C. Abadie-Type constraint qualification for mathematical programs with equilibrium constraints [J]. Journal of Optimization Theory and Applications, 2005, 124(3): 595-614.
Chen Y, Florian M. The nonlinear bilevel programming problem: formulations, regularity, and optimality conditions [J]. Optimization, 1995, 32: 193-209.
Bazaraa M S, Sherali H D, Shetty C M. Nonliner Programming: Theory and Algorithms (3rd ed.) [M]. New York: John Wiley {\& Sons, 2006.
Fujiwara O, Han S P, Mangasarian O L. Local duality of nonlinear programs [J]. SIAM Journal on Control and Optimization, 1984, 22: 162-169.
Janin R. Directional derivative of the marginal function in nonlinear programming [J]. Mathematical Programming Study, 1984, 21: 110-126.
Qi L Q, Wei Z X. On the constant positive linear dependence condition and its application to SQP methods [J]. SIAM Journal on Optimization, 2000, 10: 963-981.
Minchenko L, Stakhovsji S. On relaxed constant rank regularity condition in mathematical programming [J]. Optimization, 2011, 60(4): 429-440.
Andreani R, Haeser G, Schuverdt M, et al. A relaxed constant positive linear dependence constraint qualification and applications [J]. Mathematical Programming, 2012, 135(1-2): 255-273.
Bertsekas D P. Nonlinear Programming (2nd ed.) [M].Scientific, Belmont, 1999.
Hestenes M R. Optimization Theory: the Finite Dimensional Case [M]. New York: John Wiley {\& Sons, 1975.
Scheel H, Scholtes S. Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity [J]. Mathematics of Operations Research, 2000, 25: 1-22.
Andreani R, Martinez J M, Schuverdt M L. On the relation between constant positive linear dependence condition and quasinormality constraint qualification [J]. Journal of Optimization Theory and Applications, 2005, 125(2): 473-485.
Lu S. Relation between the constant rank and the relaxed constant rank constraint qualifications [J]. Optimization, 2012, 61(5): 555-566.
Pang J S, Fukushima M. Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints [J]. Computational Optimization and Appications, 1999, 13: 111-136.
Flegel M L, Kanzow C. On the Guignard constraint qualification for mathematical programs with equilibrium constraints [J]. Optimization, 2005, 54: 517-534.
Ye J J. Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints [J]. Journal of Mathematical Analysis and Applications, 2005, 307: 350-369.
Hoheisel T, Kanzow C, Schwartz A. Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints [J]. Mathematical Programming, 2013, 137: 257-288.
Guo L, Lin G H, Ye J J. Second order optimality conditions for mathematical programs with equilibrium constraint [J]. Journal of Optimization Theory and Applications, 2013, 158: 33-64.
Guo L, Lin G H. Notes on some constraint qualifications for mathematical programs with equilibrium constraints [J]. Journal of Optimization Theory and Applications, 2013, 156(3): 600-616.
Kanzow C, Schwartz A. Mathematical programs with equilibrium constraints: enhanced fritz john conditions, new constraint qualifications and improved exact penalty results [J]. SIAM Journal on Optimization, 2010, 20: 2730-2753.
Flegel M L, Kanzow C. A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints [J]. Optimization, 2003, 52: 277-286.
Outrata J V. Optimality conditions for a class of mathematical programs with equilibrium constraints [J]. Mathematics of Operations Research, 1999, 24: 627-644.
Outrata J V. A generalized mathematical program with equilibrium constraints [J]. SIAM Journal on Control and Optimization, 2000, 38: 1623-1638.
Flegel M L, Kanzow C. On M-stationary points for mathematical programs with equilibrium constraints [J]. Journal of Mathematical Analysis and Applications, 2005, 310: 286-302.
Flegel M L. Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints [D]. Wurzburg: University of Wurzburg, 2005.
Ye J J. Optimality conditions for optimization problems with complementarity constraints [J]. SIAM Journal on Optimization, 1999, 9: 374-387.
Andreani R, Haeser G, Schuverdt M, et al. Two new weak constraint qualifications and applications [J]. SIAM Journal on Optimization, 2012, 22(3): 1109-1135.

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Review on constraint qualifications and optimality conditions for mathematical programs with equilibrium constraints

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