Equivalent Primal and Dual Differentiable Reformulations of the Euclidean Multifacility Location Problem

As introduced earlier, the Euclidean Multifacility Location Problem (EMFLP) seeks to locate n new facilities at some points (xi, yi), i = 1, .., n in R 2 , in the presence of some m existing facilities located at specified points (aj, bj), j = 1, .., m, given certain interaction weights wij > 0 between designated pairs (i, j) of new and existing facilities in some indexpair set A NE , as well as certain interaction weights vkl > 0 between designated pairs (k, l), k < l, of new facilities themselves in some index-pair set ANN. The cost for each pair of interacting facilities is assumed to be directly proportional to the interaction weight and the Euclidean distance that separates these facilities. This problem may be mathematically stated as follows: