We consider robust two-stage optimization problems, which can be considered as a game between the decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing two-stage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. In the former, we show that a wide range of robust combinatorial problems can be decomposed into polynomially many subproblems, which in turn can often be solved in polynomial time. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where first- and second-stage adversarial costs are equal can remain solvable in polynomial time.

中文翻译：

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具有两阶段不确定性的两阶段鲁棒优化问题

我们考虑健壮的两阶段优化问题，可以将其视为决策者和对手之间的博弈。决策者修复部分解决方案后，对手从指定的不确定性集中选择一种方案。之后，决策者可以通过将部分第一阶段的解决方案完成为完整的解决方案来对这种情况做出反应。我们通过在第二个决策者阶段之后添加另一个对手阶段来扩展此经典设置，这会导致出现最小-最大-最小-最大问题，从而将两阶段设置进一步推向更一般的多阶段问题。我们关注预算不确定性集，并考虑连续和离散情况。在前一种方法中，我们表明可以将多种鲁棒的组合问题分解为多项式的许多子问题，反过来，这通常可以在多项式时间内求解。对于后者，我们证明了许多问题的NP硬度，但是请注意，第一阶段和第二阶段对抗成本相等的特殊情况可以在多项式时间内解决。