Tropical convexity in location problems

Andrei Comăneci ()
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Andrei Comăneci: Technische Universität Berlin

Mathematical Methods of Operations Research, 2024, vol. 100, issue 2, No 4, 509-534

Abstract: Abstract We investigate location problems where the optimal solution is found within the tropical convex hull of the given input points. Our initial focus is on geodesically star-convex sets, using the asymmetric tropical distance. We introduce the concept of tropically quasiconvex functions, which have sub-level sets with this shape, and are closely related to monotonic functions. Our findings demonstrate that location problems using tropically quasiconvex functions as distance measures will result in an optimal solution within the tropical convex hull of the input points. We also extend this result to cases where the input points are replaced with tropically convex sets. Finally, we explore the applications of our research in phylogenetics, highlighting the properties of consensus methods that arise from our class of location problems.

Keywords: Tropical convexity; Tropically quasiconvex function; Monotonic norms; Tropical $$L^p$$ L p -norm; Tropically convex consensus trees; 14T90; 26B25; 52A30; 90B85; 92B10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00186-024-00869-w

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