Executive Summary
This paper extends the tropical Fermat–Weber problem to include non-finite data and explores its inverse formulations. It provides linear programming solutions for these problems, building on previous work in tropical geometry and location theory. The research originates from the Naval Postgraduate School.
Why It Matters
This document, originating from the Naval Postgraduate School, contributes to advanced mathematical optimization techniques that could be applied to complex logistical or resource allocation problems relevant to defense operations, even if not explicitly stated.
Key Takeaways
- The tropical Fermat–Weber problem is extended to handle non-finite data, broadening its applicability to more complex scenarios.
Source evidence · PDF page 2
We extend previous formulations in considering non-finite data, and we investigate several variants of the inverse problem, providing linear programming formulations for their solution.
- A linear programming framework is developed to solve the extended tropical Fermat–Weber problem, including cases with non-finite coordinates.
Source evidence · PDF page 12
In this section, we extend our previous linear programming framework for solving TFW to samples where the coordinates are not limited to finite elements.
- The paper investigates inverse formulations of the tropical Fermat–Weber problem, providing linear programming solutions for determining input parameters given an optimal solution.
Source evidence · PDF page 27
We also consider inverse formulations of tropical Fermat–Weber primal and dual problems, providing some characterizations on the space of feasible inverse solutions and providing linear programming formulations to compute their solution.
Strategic Relevance
The mathematical advancements in location problems and inverse optimization could enhance logistical planning, resource deployment, and strategic positioning models for military applications, particularly in complex or uncertain environments.