The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

Abstract

Given a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.