The Robust Bilevel Selection Problem

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.