Reoptimization in Lagrangian methods for the 0–1 quadratic knapsack problem

The 0–1 quadratic knapsack problem consists of maximizing a quadratic objective function subject to a linear capacity constraint. To exactly solve large instances of this problem with a tree search algorithm (e.g., a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0–1 linear knapsack subproblems. Thus, taking this huge number of resolutions into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0–1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0–1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach.