I. Norman Katz

Facility location in the presence of forbidden regions, I: Formulation and the case of Euclidean distance with one forbidden circle

Abstract A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region R of the plane. Using the calculus of variations the shortest path between two points x , y ∉ R which does not intersect R is determined. If d( x , y ) is unconstrained distance, we denote the shortes distance along a feasible path by d ( x y ) . The constrained Weber problem is, then: given points x j ∉ R and positive weights wj, j = 1,2,…,n, find a point x ∉ R such that f( x )= Σ n j=1 d( x , x j ) is a minimum. An algorithm is formulated for the solution of this problem when d( x , y ) is Euclidean distance and R is a single circular region. Numerical results are presented.

Local convergence in Fermat's problem

The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination.If  is not a destination, then locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′ (λ ≥ 1/2, whenn=2). If  is a destination, then convergence can be either linear, quadratic or sublinear. Three numerical examples which illustrate the different possibilities are given and comparisons are made with the use of Steffensens scheme to accelerate convergence.

A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

SummaryThe argument principle is a natural and simple method to determine the number of zeros of an analytic functionf(z) in a bounded domainD. N, the number of zeros (counting multiplicities) off(z), is 1/2π times the change in Argf(z) asz moves along the closed contour σD. Since the range of Argf(z) is (−π, π] a critical point in the computational procedure is to assure that the discretization of σD, {zi,i=1, ...,M}, is such that

An Always-Convergent Numerical Scheme for a Random Locational Equilibrium Problem

|\Delta _{{\text{[z}}_i {\text{,}} {\text{z}}_{i + 1} {\text{]}}} Arg f(z)| \leqq \pi

An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations

. Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of σD lead to a completely reliable algorithm to computeN. This algorithm also treats in an elementary way the case when a zero is on or near the contour σD. Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.

The p-Version of the Finite Element Method for Problems Requiring

Abstract The problem of finding a point on the sphere S2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied. Three different metrics are considered as distances between points on S2: (A), square of Euclidean distance; (B), Euclidean distance; (C), great circle distance. Non uniqueness of minimizers is demonstrated and some pathological cases are studied. An algorithm, analogous to the Weiszfeld algorithm for the classical unconstrained Weber problem is formulated, and its convergence properties are investigated. A necessary and sufficient condition for a destination point to be a local minimizer is derived. Finally, a modified form of Steffensens acceleration is given and the results of numerical tests are presented. These results illustrate the predictions of the theory, and confirm the effectiveness of Steffensens acceleration.

C^1

A probabilistic extension of the classical Weber problem is studied. N destinations in the plane,

-Continuity

P_j ,j = 1, \cdots ,N