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.

在本文中，我们扩展了我们最近针对三维分配问题提出的纯对偶公式，以解决更通用的多维分配问题。推导了凸对偶表示，并研究了它与通常用于解决多维分配问题的拉格朗日松弛方法的关系。另外，我们讨论了对偶间隙为零的条件。还指出，拉格朗日松弛的过程本质上等同于松弛二元约束条件之一，因此需要拍卖搜索操作来恢复二元约束。此外，提出了一种基于对偶公式的数值算法以及局部搜索策略。