Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs
提出一种名为“阶段t情景支配”的新方法，通过比较各阶段不确定参数的部分排序来生成切割平面，显著加速求解风险规避的多阶段随机混合整数规划问题，并在动态背包问题中验证了计算效率提升一到两个数量级。
Abstract This paper presents a new and general approach, named “Stage- t Scenario Dominance,” to solve the risk-averse multi-stage stochastic mixed-integer programs (M-SMIPs). Given a monotonic objective function, our method derives a partial ordering of scenarios by pairwise comparing the realization of uncertain parameters at each time stage under each scenario. Specifically, we derive bounds and implications from the “Stage- t Scenario Dominance” by using the partial ordering of scenarios and solving a subset of individual scenario sub-problems up to stage t . Using these inferences, we generate new cutting planes to tackle the computational difficulty of risk-averse M-SMIPs. We also derive results on the minimum number of scenario-dominance relations generated. We demonstrate the use of this methodology on a stochastic version of the mean-conditional value-at-risk (CVaR) dynamic knapsack problem. Our computational experiments address those instances that have uncertainty, which correspond to the objective, left-hand side, and right-hand side parameters. Computational results show that our “scenario dominance"-based method can reduce the solution time for mean-risk, stochastic, multi-stage, and multi-dimensional knapsack problems with both integer and continuous variables, whose structure is similar to the mean-risk M-SMIPs, with varying risk characteristics by one-to-two orders of magnitude for varying numbers of random variables in each stage. Computational results also demonstrate that strong dominance cuts perform well for those instances with ten random variables in each stage, and ninety random variables in total. The proposed scenario dominance framework is general and can be applied to a wide range of risk-averse and risk-neutral M-SMIP problems.