Abstract
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.
Highlights
Bilevel optimization has received increasing attention in the last decades
We investigate whether—and in which cases—taking uncertainty of some problem parameters into account renders a bilevel optimization problem significantly harder, where we focus on the robust optimization approach
We have started the investigation of robust bilevel optimization by addressing the bilevel continuous knapsack problem with uncertain follower’s objective
Summary
Bilevel optimization has received increasing attention in the last decades. The aim is to model situations where certain decisions are taken by a so-called leader, but one or more followers optimize their own objective functions subject to the choices of the leader. In our investigation of the bilevel continuous knapsack problem, we only consider uncertainty in the coefficients of the follower’s objective function. This is a very natural setting in bilevel optimization, as in practice the leader often does not know the follower’s objective function precisely. One could look at uncertainty in the leader’s objective function (still from the leader’s perspective) which corresponds to the single-level robust optimization setting with uncertain objective, ignoring the bilevel structure of the leader’s problem. For many other types of uncertainty sets, the robust bilevel continuous knapsack problem turns out to be NP-hard, and the same is true even for the problem of evaluating the leader’s objective function, i.e., the adversary’s optimization problem.
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