On the complexity of robust bilevel optimization with uncertain follower's objective

Buchheim, Christoph; Henke, Dorothee; Hommelsheim, Felix

doi:10.1016/j.orl.2021.07.009

Cited by 7 publications

(1 citation statement)

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“…Among other things, it is shown that the resulting problem is still tractable under discrete or interval uncertainty-the latter case being nontrivial here, while it turns out to be NP-hard for budgeted uncertainty (which is sometimes called Gamma uncertainty) and for ellipsoidal uncertainty, among others. Complexity questions for general robust bilevel optimization problems have been settled in [3].…”

Section: Introductionmentioning

confidence: 99%

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

Buchheim

Henke

Irmai³

2022

J Optim Theory Appl

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We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader’s aim is to optimize a linear objective function in the capacity and in the follower’s solution, but with respect to different item values. We address a stochastic version of this problem where the follower’s profits are uncertain from the leader’s perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader’s objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.

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Section: Introductionmentioning

confidence: 99%