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In this paper we develop a branch-and-bound algorithm for solving a particular integer quadratic multi-knapsack problem. The problem we study is defined as the maximization of a concave separable quadratic objective function over a convex set of linear constraints and bounded integer variables. Our exact solution method is based on the computation of an upper bound and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process. We lead a numerical comparison between our method and three other existing algorithms. The approach we propose outperforms other procedures for large-scaled instances (up to 2000 variables and constraints).

In this paper we will consider the 0–1 quadratic knapsack problem (QKP). Our purpose is to show that using a linear reformulation of this problem and a standard mixed integer programming tool, it is possible to solve the QKP efficiently in terms of computation time and the size of problems considered, in comparison to existing methods. Considering a problem involving n variables, the linearization technique we propose has the advantage of adding only (n – 1) real variables and 2(n – 1) constraints. We present extensive computational results on randomly generated instances and on structured problems coming from applications. For example, the method allows us to solve randomly generated QKP instances exactly with up to 140 variables.