Stochastic mathematical programs with equilibrium constraints

带平衡约束的数学规划(MPEC)是一类用参数变分不等式或互补系统定义基本约束的优化问题。在某种程度上，MPEC是双层规划(BLP)或Stackelberg游戏的扩展。MPEC公式在最优控制(裂纹结构的最优预应力)、工程(动态刚体模型、非线性障碍问题)、经济学(Stackelberg博弈、交通规划、设施选址、定价、供应链管理)中有多种应用。从数学上讲，一般的MPEC是一个高度非凸、不可微的优化问题，它的约束条件中包含了某些组合特征。因此，MPEC对于理论和算法研究人员来说都是非常丰富的。MPEC中的基础数据是确定性的。然而，在许多应用中，一些数据可能包含随机(不确定)因素，这使得具有平衡约束的随机数学规划(SMPEC)成为可能。忽视这些随机因素可能会导致在特定市场实现的基础上做出最优决策，这是不现实的。

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.