Stochastic mathematical programs with equilibrium constraints

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Mathematical programs with equilibrium constraints (MPECs) are an optimization problem in which the essential constraints are defined by a parametric variational inequality or complementarity system. In a certain way, MPECs are an extension of the class of bilevel programs (BLP) or Stackelberg games. There are diverse applications that lend themselves to an MPEC formulation in optimal control (optimal prestress of cracked structures), engineering (dynamic rigid-body model, nonlinear obstacle problems), economics (Stackelberg games, transit planning, facility location, pricing, supply chain management). Mathematically, the general MPECs are a highly non-convex, non-differentiable optimization problem that encompasses certain combinatorial features in its constraints. Hence, MPEC is very rich for both theoretical and algorithmic researchers. The underlying data in MPEC are deterministic. However, in many applications, some data may involve stochastic (uncertain) factors and this makes a case for stochastic mathematical programs with equilibrium constraints (SMPEC). Ignoring such stochastic factors may result in an optimal decision being made on the basis of a particular market realization which is not realistic.