An efficient global algorithm for indefinite separable quadratic knapsack problems with box constraints.

The indefinite separable quadratic knapsack problem (ISQKP) with box constraints is known to be NP-hard. In this paper, we propose a new branch-and-bound algorithm based on a convex envelope relaxation that can be efficiently solved by exploiting its special dual structure. Benefiting from a new branching strategy, the complexity of the proposed algorithm is quadratic in terms of the number of variables when the number of negative eigenvalues in the objective function of ISQKP is fixed. We then improve the proposed algorithm for the case that ISQKP has symmetric structures. The improvement is achieved by constructing tight convex relaxations based on the aggregate functions. Numerical experiments on large-size instances show that the proposed algorithm is much faster than Gurobi and CPLEX. It turns out that the proposed algorithm can solve the instances of size up to three million in less than twenty seconds on average and its improved version is still very efficient for problems with symmetric structures. [ABSTRACT FROM AUTHOR]