An interior-point method for the single-facility location problem with mixed norms using a conic formulation

Abstract

We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in \({\mathbb{R}^n}\) , where each distance can be measured according to a different p-norm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to any given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.