Weiszfeld Procedure Research Articles

Application of Heuristic and Metaheuristic Algorithms in Solving Constrained Weber Problem with Feasible Region Bounded by Arcs

The continuous planar facility location problem with the connected region of feasible solutions bounded by arcs is a particular case of the constrained Weber problem. This problem is a continuous optimization problem which has a nonconvex feasible set of constraints. This paper suggests appropriate modifications of four metaheuristic algorithms which are defined with the aim of solving this type of nonconvex optimization problems. Also, a comparison of these algorithms to each other as well as to the heuristic algorithm is presented. The artificial bee colony algorithm, firefly algorithm, and their recently proposed improved versions for constrained optimization are appropriately modified and applied to the case study. The heuristic algorithm based on modified Weiszfeld procedure is also implemented for the purpose of comparison with the metaheuristic approaches. Obtained numerical results show that metaheuristic algorithms can be successfully applied to solve the instances of this problem of up to 500 constraints. Among these four algorithms, the improved version of artificial bee algorithm is the most efficient with respect to the quality of the solution, robustness, and the computational efficiency.

ALGORITHM FOR WEBER PROBLEM WITH A METRIC BASED ON THE INITIAL FARE

We introduce a non-Euclidean metric for transportation sys- tems with a dened minimum transportation cost (initial fare) and in- vestigate the continuous single-facility Weber location problem based on this metric. The proposed algorithm uses the results for solving the We- ber problem with Euclidean metric by Weiszfeld procedure as the initial point for a special local search procedure. The results of local search are then checked for optimality by calculating directional derivative of modi- ed objective functions in nite number of directions. If the local search result is not optimal then algorithm solves constrained Weber problems with Euclidean metric to obtain the nal result. An illustrative example is presented.

The geometric median on Riemannian manifolds with application to robust atlas estimation

One of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images. The study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space, which map one anatomy to another. It is now well established that representing the geometry of shapes or images in Euclidian spaces undermines our ability to represent natural variability in populations. In our previous work we have extended classical statistical analysis techniques, such as averaging, principal components analysis, and regression, to Riemannian manifolds, which are more appropriate representations for describing anatomical variability. In this paper we extend the notion of robust estimation, a well established and powerful tool in traditional statistical analysis of Euclidian data, to manifold-valued representations of anatomical variability. In particular, we extend the geometric median, a classic robust estimator of centrality for data in Euclidean spaces. We formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. In this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images. Using this approach, we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images.

Extension of the Weiszfeld procedure to a single facility minisum location model with mixed ℓ p norms

This paper presents a general mixed-norm minisum problem for locating a single facility in continuous space. It is assumed that several transportation modes exist between the new facility and a given set of fixed points (the customers), each mode being represented by a different l p norm. A simple extension of Weiszfeld’s well known iterative procedure is proposed to solve the model. Convergence properties and optimality criteria are derived, and computational results are given.

Local improvement in planar facility location using vehicle routing

In physical distribution the location of depots and vehicle routes are interdependent problems, but they are usually treated independently. Location-routing is the study of solving locational problems such that routing considerations are taken into account. We present an iterative heuristic for the location-routing problem on the plane. For each depot the Weber problem is solved using the end-points of the routes found previously as input nodes to the Weiszfeld procedure. Although the improvements found are usually small they show that it pays not to ignore the routing aspects when solving continuous location problems. Possible research avenues in continuous location-routing will also be suggested.

Finding the optimal solution to the Huff based competitive location model

In this paper we solve the gravity (Huff) model for the competitive facility location problem. We show that the generalized Weiszfeld procedure converges to a local maximum or a saddle point. We also devise a global optimization procedure that finds the optimal solution within a given accuracy. This procedure is very efficient and finds the optimal solution for 10,000 demand points in less than six minutes of computer time. We also experimented with the generalized Weiszfeld algorithm on the same set of randomly generated problems. We repeated the algorithm from 1,000 different starting solutions and the optimum was obtained at least 17 times for all problems.

The convergence of the Weiszfeld algorithm

We investigate the convergence properties of the Weiszfeld procedure when it is applied to the approximated ℓ p -norm single-facility location problem where p > 2. We show that convergence for p > 2 can be obtained by introducing a step size factor to the iterative procedure. Some numerical test results are also given.

A note on convergence in the single facility minisum location problem

The single facility minisum location problem requires finding a point in R N that minimizes a sum of weighted distances to m given points. The distance measure is typically assumed in the literature to be either Euclidean or rectangular, or the more general l p norm. Global convergence of a well-known iterative solution method named the Weiszfeld procedure has been proven under the proviso that none of the iterates coincide with a singular point of the iteration functions. The purpose of this paper is to examine the corresponding set of “bad” starting points which result in failure of the algorithm for a general l p norm. An important outcome of this analysis is that the set of bad starting points will always have a measure zero in the solution space ( R N ), thereby validating the global convergence properties of the Weiszfeld procedure for any l p norm, p ∈ [1, 2].

A note on accelerating the weiszfeld procedure

Abstract An accelerated method for achieving faster convergence of the Weiszfeld procedure used in the solution of the planar Weber problem is proposed. The acceleration is based on two consecutive iterations and assumes that the differences between successive points form a geometric series. The next iterate is the limit of this geometric series. Computational experiments are reported.

Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances

This paper considers a general form of the single facility minisum location problem (also referred to as the Fermat-Weber problem), where distances are measured by an lp norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean (p = 2) distances.

Local convergence in a generalized Fermat-Weber problem

In the Fermat-Weber problem, the location of a source point in ℝ N is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by anl p norm and the parameterp takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values ofp exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.

A note on the Weber location problem

In this note, we collect some interesting and useful results about the Weber problem. We investigate an accelerated Weiszfeld procedure which increases the step size and find a formula for the step size that empirically produces the fastest convergence rate. We also derive an estimate for the optimal cost of the system.

Solution of location problems with radial cost functions

The solution of the Cooper location problem min ▪ where r i is the radial (Euclidean) distance between the ith given location ( a i , b i and the center ( x, y) to be located is further investigated. The iterative mehtod given by Cooper (which includes the well known Weiszfeld procedure for n = 1) was previously amended using semi-intuitive arguments. In the present work a better proof is offered for the results given before. Furthermore, using the same line of argument, a broader group of problems previously mentioned by Katz and others can be efficiently solved. These are the problems min ▪ where ψ i are non-decreasing functions of the Euclidean distances. The method is also extended to solve similar problems in E K with K> 2. Apart from the theoretical account, computational experience is reported for the three dimensional Cooper problem with differet values of n. Computational results of the min ▪ which is a different member of the Katz class of problems, are also presented.