Local convexity results in a generalized Fermat-Weber problem

A generalized form of the Fermat-Weber problem requires finding a point in N to minimize a sum of nondecreasing functions of distances to m given points. In this paper, local convexity properties are investigated for the generalized problem. Sufficient conditions are derived which guarantee that the Hessian matrix of the objective function will be positive definite. The analysis also reveals that Weiszfeld-type iterative algorithms may have sublinear convergence rates, since the Hessian may only be positive semidefinite at a local minimum.