Abstract
We revisit the polyhedral projection problem. This problem has many applications, among them certain problems in global optimisation, polyhedral calculus, problems encountered in information theory and financial mathematics. In particular, it has been shown recently that polyhedral projection problems are equivalent to vector linear programmes (which contain multiple objective linear programmes as a sub-class). In this article, we develop a novel solution concept which provides more detailed insights into the structure of the projected polyhedron by taking its lineality space into account. We explore the relationship of our new solution concept to a previous one. We extend the problem class of vector linear programmes by using pre-orders instead of partial orders. We then show that solutions (according to the lattice approach) to such vector linear programmes can be derived by solving a related polyhedral projection problem.
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Notes
In this article, a polyhedron is always meant to be a convex polyhedron.
We slightly adapt the definition here. In Löhne and Weißing (2016), zero is not a valid direction and \(\smash {X^\mathrm {dir}}\) is empty for bounded polyhedra (
).
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Weißing, B. The polyhedral projection problem. Math Meth Oper Res 91, 55–72 (2020). https://doi.org/10.1007/s00186-019-00677-7
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DOI: https://doi.org/10.1007/s00186-019-00677-7