A Polynomial Algorithm for a Continuous Bilevel Knapsack Problem

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity O(n). Keywords— Bilevel programming, Continuous knapsack problem, Polynomial time. 1 Model Recently, a number of papers have been devoted to bilevel programs involving integer-valued knapsacks at the lower level [2]–[6],[9], all of them NP-hard. A question that arises naturally is the complexity of the continuous variants of these problems. More precisely, we consider a continuous bilevel knapsack problem with interdiction constraints that consists in a relaxation of the problem analyzed by Caprara et al.[4], where the goal of the leader is to minimize the value of the follower’s knapsack, through the interdiction of a subset of items. Its mathematical formulation is (CBK) min (x,y)∈[0,1]×[0,1] n ∑