In bilevel optimization problems, a leader and a follower make their decisions in a hierarchy, and both decisions may 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 from their own item set, 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 two players' item sets 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 optimal 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，领导者与跟随者以层级结构进行决策，双方的决策可能相互影响。通常假设双方均完全知晓对方的数据。在更现实的模型中，不确定性可被量化，例如采用鲁棒优化方法：我们假设领导者无法精确获知跟随者的目标函数，仅知其属于某个不确定集，其目标是在对应情景的最坏情况下进行优化。由此产生的问题是：在这种不确定性的额外复杂性下，双层优化的复杂度将如何变化？我们通过研究一个简单的双层问题向回答该问题迈进一步。在双层选择问题中，领导者与跟随者分别从各自的物品集中选择若干物品，同时给定需选择物品的总数，双方根据不同的物品成本集最小化所选物品的总成本。我们证明BSP可在多项式时间内求解，进而研究其鲁棒版本。若两位决策者的物品集互不相交，针对多种不确定集类型该问题仍可在多项式时间内求解。否则，我们证明鲁棒BSP是NP难问题，并提出一种2-近似算法及精确指数时间求解方法。此外，我们研究了BSP的若干变体，其中一位或两位决策者采用连续决策。某变体引出了双层优化问题最优值可能无法达到的示例。针对全变量连续的鲁棒连续BSP，我们还为离散不相关不确定性情境开发了新方法，该方法为鲁棒连续BSP提供了多项式时间算法，并为鲁棒双层连续背包问题提供了伪多项式时间算法。