Large-scale mixed-integer linear programming (MILP) problems, such as those from two-stage stochastic programming, usually have a decomposable structure that can be exploited to design efficient optimization methods. Classical Benders decomposition can solve MILPs with weak linking constraints (which are decomposable when linking variables are fixed) but not strong linking constraints (which are not decomposable even when linking variables are fixed). In this paper, we first propose a new rigorous bilevel decomposition strategy for solving MILPs with strong and weak linking constraints, then extend a recently developed cross decomposition method based on this strategy. We also show how to apply the extended cross decomposition method to two-stage stochastic programming problems with conditional-value-at-risk (CVaR) constraints. In the case studies, we demonstrate the significant computational advantage of the proposed extended cross decomposition method as well as the benefit of including CVaR constraints in stochastic programming.

大规模混合整数线性规划（MILP）问题（例如来自两阶段随机规划的问题）通常具有可分解的结构，可用于设计有效的优化方法。经典的Benders分解可以解决具有弱链接约束（固定链接变量时可分解）的MILP，但不能解决强链接约束（即使固定链接变量时也无法分解）的MILP。在本文中，我们首先提出一种新的严格的双层分解策略，用于解决具有强链接约束和弱链接约束的MILP，然后扩展基于该策略的最新开发的交叉分解方法。我们还展示了如何将扩展交叉分解方法应用于具有条件值风险（CVaR）约束的两阶段随机规划问题。在案例研究中