We consider the problems of determining the metric dimension and the minimum cardinality of doubly resolving sets in \(n\)-cubes. Most heuristics developed for these two NP-hard problems use a function that counts the number of pairs of vertices that are not (doubly) resolved by a given subset of vertices, which requires an exponential number of distance evaluations, with respect to \(n\). We show that it is possible to determine whether a set of vertices (doubly) resolves the \(n\)-cube by solving an integer program with \(O(n)\) variables and\(O(n)\) constraints. We then demonstrate that small resolving and doubly resolving sets can easily be determined by solving a series of such integer programs within a swapping algorithm. Results are given for hypercubes having up to a quarter of a billion vertices, and new upper bounds are reported.