In bilevel optimization problems, a leader and a follower make their decisions in a hierarchy, and both decisions influence each other. Usually one assumes that both players have full knowledge also of the other player's data. In a more realistic model, uncertainty can be quantified, e.g., using the robust optimization approach: We assume that the leader does not know the follower's objective precisely, but only up to some uncertainty set, and her aim is to optimize the worst case of the corresponding scenarios. Now the question arises how the complexity of bilevel optimization changes under the additional complications of this uncertainty. We make a further step towards answering this question by examining an easy bilevel problem. In the Bilevel Selection Problem (BSP), the leader and the follower each select some items, while a common number of items to select in total is given, and each player minimizes the total costs of the selected items, according to different sets of item costs. We show that the BSP can be solved in polynomial time and then investigate its robust version. If the item sets controlled by the players are disjoint, it can still be solved in polynomial time for several types of uncertainty sets. Otherwise, we show that the Robust BSP is NP-hard and present a 2-approximation algorithm and exact exponential-time approaches. Furthermore, we investigate variants of the BSP where one or both of the two players take a continuous decision. One variant leads to an example of a bilevel optimization problem whose optimum value may not be attained. For the Robust Continuous BSP, where all variables are continuous, we also develop a new approach for the setting of discrete uncorrelated uncertainty, which gives a polynomial-time algorithm for the Robust Continuous BSP and a pseudopolynomial-time algorithm for the Robust Bilevel Continuous Knapsack Problem.

翻译：在双层优化问题中，领导者与跟随者按层级顺序做出决策，且两者决策相互影响。通常假设双方均完全知晓对方的决策数据。在更贴近实际的模型中，不确定性可通过鲁棒优化方法加以量化：假设领导者无法精确获知跟随者的目标函数，仅能知晓其存在于某个不确定集合中，其目标是最优化对应情景下的最坏情况。此时需探究此不确定性增加后对双层优化复杂性的影响。我们通过分析一个简单的双层问题，向解答此问题迈出进一步步伐。在双层选择问题中，领导者与跟随者各自选择若干物品，且存在共同的总选择数量约束，双方根据不同的物品成本集合最小化所选物品的总成本。我们证明该问题可在多项式时间内求解，并进一步研究其鲁棒版本。若双方控制的物品集互不相交，则对于多类不确定集合，该问题仍可在多项式时间内求解。反之，我们证明鲁棒双层选择问题是NP难问题，并提出一个2-近似算法及精确指数时间方法。此外，我们研究了双方或其中一方采用连续决策的双层选择问题变体。其中一个变体提供了最优值可能无法达到的双层优化问题实例。对于所有变量均为连续的鲁棒连续双层选择问题，我们针对离散非关联不确定性设定提出新方法，该方法可提供鲁棒连续双层选择问题的多项式时间算法，以及鲁棒双层连续背包问题的伪多项式时间算法。