The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

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.

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

“…Moreover, the constructions in the proofs of Theorem 2 and 3 show that all stated hardness results still hold when δ = 0. One can show that, since all follower's item values in these constructions are positive, the hardness result also holds when assuming d ≥ 0 and δ > 0; see [2].…”

“…Regarding the bilevel continuous knapsack problem under uncertainty, the robust optimization approach has been investigated in depth in [2]. It is assumed that the vector of follower's item values is unknown to the leader.…”

On the Complexity of Robust Bilevel Optimization With Uncertain Follower's Objective

“…The result of Theorem 1 does not only hold for interval uncertainty sets, but also for many other uncertainty sets U where each entry of c can be chosen to be positive or negative independently of each other. In particular, the proof of Theorem 1 can be easily adapted for the case of uncorrelated discrete uncertainty, as considered in [5].…”

“…Addressing similar complexity questions, Buchheim and Henke [5] investigate a bilevel continuous knapsack problem where the leader controls the continuous capacity of the knapsack and, as in (R), the follower's profits are uncertain. They show that (the analoga of) both (F) and (P) can be solved in polynomial time in this case, while (A) and (R) turn out to be NP-hard for, e.g., budgeted and ellipsoidal uncertainty.…”

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

“…In the last years, algorithmic research on bilevel optimization focused on more and more complicated lower-level problems such as mixed-integer linear models (Fischetti et al 2017(Fischetti et al , 2018, nonlinear but still convex models (Kleinert, Grimm, et al 2021), or problems in the lower level with uncertain data (Buchheim and Henke 2022;Burtscheidt and Claus 2020). When it comes to the situation of a lowerlevel problem with continuous nonlinearities there is not too much literature-in particular in comparison to the case in which the lower-level problem is convex; see, e.g., Kleniati and Adjiman (2011, 2014a,b, 2015, Mitsos (2010), Mitsos et al (2008), , Paulavičius, Gao, et al (2020), and Paulavičius et al (2016).…”