Geodesic
A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints
Kybernetika, Tome 46 (2010) no. 3, pp. 423-434 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.
In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.
Classification : 49J53, 90C30, 90C31, 90C47
Keywords: mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone 
@article{KYB_2010_46_3_a7,
     author = {Henrion, Ren\'e and Outrata, Ji\v{r}{\'\i} and Surowiec, Thomas},
     title = {A note on the relation between strong and {M-stationarity} for a class of mathematical programs with equilibrium constraints},
     journal = {Kybernetika},
     pages = {423--434},
     year = {2010},
     volume = {46},
     number = {3},
     mrnumber = {2676080},
     zbl = {1225.90125},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a7/}
}
TY  - JOUR
AU  - Henrion, René
AU  - Outrata, Jiří
AU  - Surowiec, Thomas
TI  - A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints
JO  - Kybernetika
PY  - 2010
SP  - 423
EP  - 434
VL  - 46
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a7/
LA  - en
ID  - KYB_2010_46_3_a7
ER  - 
%0 Journal Article
%A Henrion, René
%A Outrata, Jiří
%A Surowiec, Thomas
%T A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints
%J Kybernetika
%D 2010
%P 423-434
%V 46
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a7/
%G en
%F KYB_2010_46_3_a7
Henrion, René; Outrata, Jiří; Surowiec, Thomas. A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints. Kybernetika, Tome 46 (2010) no. 3, pp. 423-434. http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a7/

[1] Dontchev, A. L., Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6 (1996), 1087–1105. | DOI | MR | Zbl

[2] Dontchev, A. L., Rockafellar, R. T.: Ample parameterization of variational inclusions. SIAM J. Optim. 12 (2001), 170–187. | DOI | MR | Zbl

[3] Henrion, R., Jourani, A., Outrata, J. V.: On the calmness of a class of multifunctions. SIAM J. Optim. 13 (2002), 603–618. | DOI | MR | Zbl

[4] Henrion, R., Römisch, W.: On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling. Appl. Math. 52 (2007), 473–494. | DOI | MR

[5] Henrion, R., Outrata, J. V., Surowiec, T.: On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal. 71 (2009), 1213–1226. | DOI | MR

[6] Henrion, R., Outrata, J. V., Surowiec, T.: Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market. Weierstraß-Institute of Applied Analysis and Stochastics, Preprint No. 1433 (2009) and submitted. | MR

[7] Henrion, R., Mordukhovich, B. S., Nam, N. M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim., to appear. | MR

[8] Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge 1996. | MR | Zbl

[9] Mordukhovich, B. S.: Approximation Methods in Problems of Optimization and Control (in Russian). Nauka, Moscow 1988. | MR

[10] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. Vol. 1: Basic Theory. Springer, Berlin 2006.

[11] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. Vol. 2: Applications. Springer, Berlin 2006. | MR

[12] Flegel, M. L., Kanzow, C., Outrata, J. V.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15 (2007), 139–162. | DOI | MR | Zbl

[13] Outrata, J. V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht 1998. | MR

[14] Pang, J.-S., Fukushima, M.: Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13 (1999), 111–136. | DOI | MR

[15] Robinson, S. M.: Some continuity properties of polyhedral multifunctions. Math. Program. Studies 14 (1976), 206–214. | DOI | MR | Zbl

[16] Robinson, S. M.: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43–62. | DOI | MR | Zbl

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

[18] Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity. Math. Oper. Res. 25 (2000), 1–22. | DOI | MR

[19] Surowiec, T.: Explicit Stationarity Conditions and Solution Characterization for Equilibrium Problems with Equilibrium Constraints. Doctoral Thesis, Humboldt University, Berlin 2009.