跟随者目标系数不确定的鲁棒二层连续背包问题
The robust bilevel continuous knapsack problem with uncertain coefficients in the follower's objective
摘要：
我们考虑一个双层连续背包问题，其中领导者控制背包的容量，追随者根据自己的利润选择最优包装，这可能不同于领导者。对于这个双层问题，我们以一种自然的方式添加了不确定性，假设领导者对追随者的问题没有充分的了解。更确切地说，采用鲁棒优化方法，并假设追随者的利润属于一个给定的不确定性集，我们的目标是计算一个解决方案，优化最坏情况下的追随者的反应，从领导者的角度来看。通过研究这个问题的复杂性，就不同类型的不确定性集，我们更好地理解双层优化和鲁棒组合优化的组合迈出了第一步。我们发现，这个问题可以解决在多项式时间离散和区间的不确定性，但同样的问题成为NP-困难时，每个系数可以独立地假设只有有限数量的值。特别是，这表明，取代不确定性集的凸包可能会改变问题显着，在经典的单级鲁棒优化的情况下。对于一般的多面体不确定性，这个问题再次被证明是NP困难的，即使在不相关的情况下，椭球不确定性也是如此。所有提出的硬度结果已经适用于领导者的目标函数的评价。
英文摘要：
We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower's problem. More precisely, adopting the robust optimization approach and assuming that the follower's profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower's reaction from the leader's perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader's objective function.
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