Lagrangean methods for the 0–1 Quadratic Knapsack

It is well known that the Lagrangean decomposition provides better bounds than the Lagrangean relaxation does. Nevertheless, the Lagrangean decomposition bound is harder to compute than the Lagrangean relaxation bound. Thus, one might wonder what is the best Lagrangean method to use in a branch-and-bound algorithm. In this paper, we give an answer to such a question for the 0–1 Quadratic Knapsack Problem. We first study the Lagrangean decomposition for this problem and give new necessary optimality conditions for the dual problem which allow us to elaborate a heuristic method for solving the Lagrangean decomposition duel problem. We then introduce this method in Chaillou–Hansen–Mahieu's branch-and-bound algorithm where upper bounds were computed by Lagrangean relaxation.