Stochastic mathematical programs with equilibrium constraints

具有平衡约束的数学规划（MPECs）是一个优化问题，其中的基本约束由参数变分不等式或互补系统定义。在某种程度上，MPEC是双层规划（BLP）或Stackelberg游戏类的扩展。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（Stackelberg博弈、运输规划、设施选址、定价、供应链管理）等领域有着广泛的应用。 在数学上，一般的MPEC是一个高度非凸的，不可微的优化问题，在其约束中包含某些组合特征。因此，MPEC是非常丰富的理论和算法的研究人员。 MPEC中的基础数据是确定性的。然而，在许多应用中，一些数据可能涉及随机（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。忽略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佳决策，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.