A dual approach to multi-dimensional assignment problems

In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency.