Some properties of regularization and penalization schemes for MPECs. (English) Zbl 1097.90054

Summary: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. The existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.


90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
91A40 Other game-theoretic models


Full Text: DOI


[1] DOI: 10.1080/02331939508844048 · doi:10.1080/02331939508844048
[2] DOI: 10.1137/S1052623493257344 · Zbl 0873.49018 · doi:10.1137/S1052623493257344
[3] DOI: 10.1137/S1052623499361233 · Zbl 1010.90086 · doi:10.1137/S1052623499361233
[4] Bonnans JF, Perturbation Analysis of Optimization Problems (2000)
[5] Fukushima M, Ill-posed Variational Problems and Regularization Techniques 447 pp 99– (1999)
[6] DOI: 10.1007/s101070050048 · Zbl 0959.65079 · doi:10.1007/s101070050048
[7] Hu X, Judge Institute of Management Sciences, University of Cambridge (2003)
[8] Huang XX, Department of Applied Mathematics, The Hong Kong Polytechnic University (2001)
[9] Lin G, Department of Applied Mathematics and Physics, Kyoto University (2002)
[10] Anitescu M, Mathematics and Computer Science Division, Argonne National Laboratory
[11] DOI: 10.1023/A:1008696504163 · Zbl 1040.90500 · doi:10.1023/A:1008696504163
[12] Luo Z, Mathematical Programs with Equilibrium Constraints, Cambridge University Press (1996)
[13] DOI: 10.1023/A:1022945316191 · Zbl 1038.90100 · doi:10.1023/A:1022945316191
[14] DOI: 10.1137/S1052623499363232 · Zbl 1005.65064 · doi:10.1137/S1052623499363232
[15] DOI: 10.1023/A:1011226232107 · Zbl 1049.90125 · doi:10.1023/A:1011226232107
[16] DOI: 10.1137/S0363012996306121 · Zbl 0922.90128 · doi:10.1137/S0363012996306121
[17] Liu X, Singapore-MIT alliance, National University of Singapore (2003)
[18] Fletcher R, Department of Mathematics, University of Dundee (2002)
[19] de Miguel A, London Business School (2003)
[20] Raghunathan AU, Department of Chemical Engineering, Carnegie Mellon University (2003)
[21] Benson HY, Department of Operations Research and Financial Engineering, Princeton University (2002)
[22] Leyffer S MacMPEC AMPL collection of mathematical programs with equilibrium constraints 2000 http://www.mcs.anl.gov/leyffer/MacMPEC
[23] Izmailov AF Optimization problems with complementarity constraints: Regularity, optimality conditions, and sensitivity 2004 To appear inComputational Mathematics and Mathematical Physics
[24] Luo Z, Multilevel Optimization: Algorithms, Complexity and Applications, Kluwer Academic Publishers pp pp. 209–229– (1998)
[25] DOI: 10.1287/moor. · Zbl 1073.90557 · doi:10.1287/moor.
[26] DOI: 10.1007/BF01593777 · Zbl 0354.90075 · doi:10.1007/BF01593777
[27] DOI: 10.1007/BF01582096 · Zbl 0398.90109 · doi:10.1007/BF01582096
[28] Robinson SM, Mathematical Programming Study 19 pp 200– (1982) · Zbl 0495.90077 · doi:10.1007/BFb0120989
[29] Hu X, Department of Mathematics and Statistics, The University of Melbourne (2003)
[30] Ralph D, Computer Sciences Department, University of Wisconsin-Madison (2003)
[31] DOI: 10.1137/0713043 · Zbl 0347.90050 · doi:10.1137/0713043
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