Abstract:Abstract:We consider the p-norm multi-facility minisum location problem with linear and distance constraints, and develop the Lagrangian dual formulation for this problem. The model that we consider represents the most general location model in which the dual formulation is not found in the literature. We find that, because of its linear objective function and less number of variables, the Lagrangian dual is more useful. Additionally, the dual formulation eliminates the differentiability problem in the primal … Show more
“…However, useful bounds on the optimal value of (P) can be found by taking Euclidean distances, i.e., approximating the matrices {Q k } by the identity matrix. (d) For other results on duality in multi-facility location problems see [6], [8] and their references.…”
Given a dataset D partitioned in clusters, the joint distance function (JDF) J(x) at any point x is the harmonic mean of the distances of x from the cluster centers. The JDF is a continuous function capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. The JDF of the whole dataset, J(D) := {J(x) : x ∈ D}, is a measure of the classifiability of D, and can be used to determine the "right" number of clusters for D. A duality theory for the JDF of a dataset is given, in analogy with Kuhn's geometric duality theory for the Fermat-Weber location problem.
“…However, useful bounds on the optimal value of (P) can be found by taking Euclidean distances, i.e., approximating the matrices {Q k } by the identity matrix. (d) For other results on duality in multi-facility location problems see [6], [8] and their references.…”
Given a dataset D partitioned in clusters, the joint distance function (JDF) J(x) at any point x is the harmonic mean of the distances of x from the cluster centers. The JDF is a continuous function capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. The JDF of the whole dataset, J(D) := {J(x) : x ∈ D}, is a measure of the classifiability of D, and can be used to determine the "right" number of clusters for D. A duality theory for the JDF of a dataset is given, in analogy with Kuhn's geometric duality theory for the Fermat-Weber location problem.
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