Euclidean Multifacility Location Research Articles

Solving Euclidean multifacility location problems under circular area constraints using Rprop

The present work considers multifacility location problems with circular area constraints having interactions between sources and destinations. A detailed literature survey reveals that a little attention has been paid to problem involving area constraints. Mathematical formulation and the analytical solutions have been obtained by using Kuhn-Tucker theory. The mathematical solution procedure is very complex and time consuming. Hence, an attempt has been made to get the solution of a complex, constrained multifacility location problem using artificial neural networks (ANN). With the help of numerical examples, it has been established that within the acceptable limits resilient back-propagation (Rprop) model compares well with those obtained through analytical method.

The solution of the EMFL problem with two new facilities in a quadrangle

This paper is concerned with the analytical solution of the EMFL (Euclidean multifacility location) problem with two new facilities and four existing facilities. In Section 1, the optimality conditions for a general EMFL problem are summarized in the form presented in [1]. In Section 2, they are applied to the considered problem, in order to locate the new facilities and to partition the space of the weights (for a given set of existing facilities) into regions with the same type of solution. However, it is pointed out that a complete solution can be obtained only in particular cases.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush-Kuhn-Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

On the solution of a class of location problems. A sample problem

In Section 2 of this paper the optimality conditions for the EMFL (Euclidean Multi-Facility Location) minisum problem are shown to be a useful tool for finding the analytical solution of many simple problems. In Section 3 the problem of connecting by means of two new facilities the vertices of an isoceles triangle is completely solved.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush–Kuhn–Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications

In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an $\epsilon$-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using {\em Gaussian elimination on leaves of a tree}, we present an algorithm which computes an $\epsilon$-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in $O(N \sqrt{N}(\log(\bar c/\epsilon)+\log N))$ arithmetic operations where $\bar c$ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N2) arithmetic operations under certain conditions.

A polynomial time dual algorithm for the Euclidean multifacility location problem

The Euclidean multi-facility location (EMFL) problem is one of locating new facilities in the Euclidean space with respect to existing facilities. The problem consists of finding locations of new facilities which will minimize a total cost function which consists of a sum of costs directly proportional to the Euclidean distances between the new facilities, and costs directly proportional to the Euclidean distances between new and existing facilities. In this paper, it is shown that the dual of EMFL proposed by Francis and Cabot is equivalent to the maximization of a linear function subject to convex quadratic inequality constraints and therefore can be solved in polynomial time by interior point methods. We also establish a theorem on the duality gap and present a procedure for recovering the primal solution from an interior dual feasible solution.

On the Convergence of a Hyperboloid Approximation Procedure for the Perturbed Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple closed-form iterative algorithm in 1937. Later, numerous authors proved that it is a convergent descent algorithm. In 1973, J. Eyster, J. White and W. Wierwille extended Weiszfeld's idea and proposed a Hyperboloid Approximation Procedure (HAP) for solving the Euclidean multifacility location problem. They believed, based on considerable computational experience, that the HAP always converges. In 1977, Ostresh proved that the HAP is a descent algorithm under certain conditions. In 1981, Morris proved that a variant of the HAP always converges. However, no convergence proof for the original HAP has ever been given. In this paper, we prove that the HAP is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point.

On the Convergence of Miehle's Algorithm for the Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple iterative algorithm in 1937. Later, it was proved by numerous authors that it is a convergent descent algorithm. W. Miehle extended Weiszfeld's algorithm to solve the Euclidean multifacility location problem. Then, L. M. Ostresh proved that Miehle's algorithm is a descent algorithm. Recently, F. Rado modified Miehle's algorithm and provided several sets of sufficient conditions for the modified algorithm to converge. He also indicated that the convergence of Miehle's algorithm was an open problem. In this paper, the relationship between Miehle's multifacility location algorithm and Weiszfeld's single facility location algorithm is analyzed. Counterexamples show that Miehle's algorithm may converge to nonoptimal points for well structured problems.

A fixed point algorithm for solving the euclidean multifacility location problem in a tree

The optimality conditions for the euclidean multifacility location problem with m new facilities in a tree are stated as optimality conditions of a certain set of m Weber problems. An algorithm is presented which solves the EMFL problem by iteratively solving such Weber problems.

The Euclidean Multifacility Location Problem

To solve the Euclidean multifacility location problem, W. Miehle extended the Weiszfeld algorithm from one to several facilities. L. M. Ostresh established that the objective function for the iterates of Miehle's algorithm decreases in value, but his results do not give a convergence proof. In this paper we give a new extension of the Weiszfeld algorithm and by generalizing the Kuhn-Ostresh convergence theorem, prove separate results for convergence and for optimality, both under mild hypotheses.

A note on optimality conditions for the Euclidean. Multifacility location problem

Thekey problem of the Euclidean multifacility location (EMFL) problem is to decide whether a givendead point is optimal. If it is not optimal, we wish to compute a descent direction. This paper extends the optimality conditions of Calamai and Conn and Overton to the case when the rows of the active constraints matrix are linearly dependent. We show that linear dependence occurs wheneverG, the graph of the coinciding facilities, has a cycle. In this case the key problem is formulated as a linear least squares problem with bounds on the Euclidean norms of certain subvectors.