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 capacity of and follower chooses an optimal packing according to his own profits, which may differ from those leader. To this problem, we add uncertainty in natural way, assuming that does not have full knowledge about follower’s problem. More precisely, adopting robust optimization approach profits belong given set, our aim is compute solution optimizes worst-case reaction leader’s perspective. By investigating complexity with respect different types sets, make first steps towards better understanding combination combinatorial optimization. show can be solved polynomial time for both discrete interval uncertainty, but same becomes NP-hard when each coefficient independently assume only finite number values. In particular, demonstrates replacing sets by their convex hulls change significantly, contrast situation classical single-level For general polytopal again turns out NP-hard, true ellipsoidal even uncorrelated case. All presented hardness results already apply evaluation objective function.