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

A probabilistic extension of the classical Weber problem is studied. N destinations in the plane, Pⱼ ,j = 1, ⋯ ,N, are given as random variables with specified probability density functions, and the problem is to find the location of the point P which minimizes the expected sum of the Euclidean distances PPⱼ ̄ . Under mild assumptions on the density functions, the objective function is shown to be strictly convex and the minimum unique. An iterative scheme for finding P is shown to be a descent method which is globally convergent, and the iteration is shown to be locally linear. Finally, numerical examples using bivariate normal density functions are given.