Characterizing Sets of Lower Bounds: a Hidden Convexity Result

This study addresses sets of lower bounds in a vector space ordered by a convex cone. It is easy to see that every set of lower bounds is downward (lower?), bounded from above, with the further property that it contains the supremum of any of its subsets which admits one. Our main result proves that these conditions are also sufficient, if the ordering cone is polyhedral. We provide other characterizations and properties of sets of lower bounds in primal and dual terms and show by means of simple counterexamples that such results fail when the polyhedrality assumption is dropped.