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

Abstract

The 0-1 quadratic knapsack problem consists in maximizing a quadratic objective function subject to a linear capacity constraint. To solve exactly 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 solvings 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.

LP relaxation and dynamic programming enhancing VNS for the multiple knapsack problem with setup

“…For example, Toth (1980) and Horowitz and Sahni (1974) presented improved DP algorithms for solving KP. Plateau and Elkihel (1985) and Chajakis and Guignard (1994) proposed hybrid approaches combining DP with an enumerative scheme for KP, and Pisinger (1994) proposed a minimal algorithm based on DP for KP. We inspired our DP approach from Bellman (1957), and tailored it to solve $$\textit{KP}[{Y}_{t}]$$.…”