Stochastic Mathematical Programs with Equilibrium Constraints

Michael Patriksson, Department of Mathematics, Chalmers University 
of Technology, Gothenburg, Sweden

Laura Wynter, PRISM, Computer Science Department, Université de
Versailles, Versailles, France

ABSTRACT:

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.

KEY WORDS: Bilevel Programming, Variational Inequality Problems, 
Stochastic Programming, Existence of Solutions