Tropical convexity in location problems

We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex functions whose sub-level sets have this shape. The latter are related to monotonic functions. Then we show that location problems whose distances are measured by tropically quasiconvex functions as before give an optimum in the tropical convex hull of the input points. We also show that a similar result holds if we replace the input points by tropically convex sets. Finally, we focus on applications to phylogenetics presenting properties of consensus methods arising from our class of location problems.Comment: 19 pages, 3 figure

Breakdown points of Fermat-Weber problems under gauge distances

We compute the robustness of Fermat--Weber points with respect to any finite gauge. We show a breakdown point of $1/(1+\sigma)$ where $\sigma$ is the asymmetry measure of the gauge. We obtain quantitative results indicating how far a corrupted Fermat--Weber point can lie from the true value in terms of the original sample and the size of the corrupted part. If the distance from the true value depends only on the original sample, then we call the gauge 'uniformly robust'. We show that polyhedral gauges are uniformly robust, but locally strictly convex norms are not.Comment: 19 pages, 4 figure

Hamilton-Jacobi scaling limits of Pareto peeling in 2D

Pareto hull peeling is a discrete algorithm, generalizing convex hull peeling, for sorting points in Euclidean space. We prove that Pareto peeling of a random point set in two dimensions has a scaling limit described by a first-order Hamilton-Jacobi equation and give an explicit formula for the limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts with convex peeling, which converges to curvature flow. The proof involves direct geometric manipulations in the same spirit as Calder (2016).Comment: 50 pages, 18 figures; v2 improves exposition and extends main theorem to cover any norm in R^

Dominating Sets for Convex Functions with some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of Linear Regression and Location

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop