Patriksson, Michael; Wynter, Laura Stochastic mathematical programs with equilibrium constraints. (English) Zbl 0937.90076 Oper. Res. Lett. 25, No. 4, 159-167 (1999). Summary: We introduce stochastic mathematical programs with equilibrium constraints (SMPEC), which generalize MPEC models by explicitly incorporating possible uncertainties in the problem data to obtain robust solutions to hierarchical problems. For this problem, we establish results on the existence of solutions, and on the convexity and directional differentiability of the implicit upper-level objective function, both for continuously and discretely distributed probability distributions. In so doing, we establish links between SMPEC models and two-stage stochastic programs with recourse. We also discuss basic parallel iterative algorithms for discretely distributed SMPEC problems. Cited in 1 ReviewCited in 54 Documents MSC: 90C15 Stochastic programming 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities Keywords:bilevel programming; variational inequality problems; stochastic programming; existence of solutions PDFBibTeX XMLCite \textit{M. Patriksson} and \textit{L. Wynter}, Oper. Res. Lett. 25, No. 4, 159--167 (1999; Zbl 0937.90076) Full Text: DOI References: [4] Facchinei, F.; Jiang, H.; Qi, L., A smoothing method for mathematical programs with equilibrium constraints, Math. Programming, 85, 107-134 (1999) · Zbl 0959.65079 [5] Facchinei, F.; Soares, J., A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7, 225-247 (1997) · Zbl 0873.90096 [7] Geoffrion, A. M., Elements of large-scale mathematical programming, Management Sci., 16, 652-691 (1970) · Zbl 0209.22801 [8] Hansen, P.; Jaumard, B.; Savard, G., New branch-and-bound rules for linear bilevel programming, SIAM J. Scientific Statist. Comput., 13, 1194-1217 (1992) · Zbl 0760.65063 [9] Harker, P. T.; Choi, S.-C., A penalty function approach for mathematical programs with variational inequality constraints, Inform. Decision Technol., 17, 41-50 (1991) · Zbl 0732.90075 [10] Higle, J. L.; Sen, S., Stochastic decomposition: An algorithm for two stage linear programs with recourse, Math. Oper. Res., 16, 650-669 (1991) · Zbl 0746.90045 [11] Hogan, W. W., Directional derivatives for extremal-value functions with applications to the completely convex case, Oper. Res., 21, 188-209 (1973) · Zbl 0278.90062 [12] Larsson, T.; Patriksson, M., A class of gap functions for variational inequalities, Math. Programming, 64, 53-79 (1994) · Zbl 0819.65101 [14] Marcotte, P., Network design problem with congestion effects: A case of bilevel programming, Math. Programming, 34, 142-162 (1986) · Zbl 0604.90053 [15] Marcotte, P.; Zhu, D., Exact and inexact penalty methods for the generalized bilevel programming problem, Math. Programming, 74, 141-157 (1996) · Zbl 0855.90120 [17] Outrata, J.; Zowe, J., A numerical approach to optimization problems with variational inequality constraints, Math. Programming, 68, 105-130 (1995) · Zbl 0835.90093 [18] Robinson, S. M., Strongly regular generalized equations, Math. Oper. Res., 5, 43-62 (1980) · Zbl 0437.90094 [19] Robinson, S. M., An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res., 16, 282-309 (1991) · Zbl 0746.46039 [20] Rockafellar, R. T.; Wets, R. J.-B., A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming, Math. Programming Study, 28, 63-93 (1986) · Zbl 0599.90090 [23] Zhang, R., Problems of hierarchical optimization in finite dimension, SIAM J. Optim., 4, 521-536 (1994) · Zbl 0819.90107 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.