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Keywords:
mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone
mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone
Summary:
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.
References:
[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 10.1137/S1052623495284029 | MR 1416530 | Zbl 0899.49004
[2] Dontchev, A. L., Rockafellar, R. T.: Ample parameterization of variational inclusions. SIAM J. Optim. 12 (2001), 170–187. DOI 10.1137/S1052623400371016 | MR 1870590 | Zbl 1008.49009
[3] Henrion, R., Jourani, A., Outrata, J. V.: On the calmness of a class of multifunctions. SIAM J. Optim. 13 (2002), 603–618. DOI 10.1137/S1052623401395553 | MR 1951037 | Zbl 1028.49018
[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 10.1007/s10492-007-0028-z | MR 2357576
[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 10.1016/j.na.2008.11.089 | MR 2527541
[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 2527541
[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 2650845
[8] Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge 1996. MR 1419501 | Zbl 1139.90003
[9] Mordukhovich, B. S.: Approximation Methods in Problems of Optimization and Control (in Russian). Nauka, Moscow 1988. MR 0945143
[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 2191745
[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 10.1007/s11228-006-0033-5 | MR 2321948 | Zbl 1149.90143
[13] Outrata, J. V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht 1998. MR 1641213
[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 10.1023/A:1008656806889 | MR 1704116
[15] Robinson, S. M.: Some continuity properties of polyhedral multifunctions. Math. Program. Studies 14 (1976), 206–214. DOI 10.1007/BFb0120929 | MR 0600130 | Zbl 0449.90090
[16] Robinson, S. M.: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43–62. DOI 10.1287/moor.5.1.43 | MR 0561153 | Zbl 0437.90094
[17] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis. Springer, Berlin 1998. MR 1491362 | Zbl 0888.49001
[18] Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity. Math. Oper. Res. 25 (2000), 1–22. DOI 10.1287/moor.25.1.1.15213 | MR 1854317
[19] Surowiec, T.: Explicit Stationarity Conditions and Solution Characterization for Equilibrium Problems with Equilibrium Constraints. Doctoral Thesis, Humboldt University, Berlin 2009.
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