Optimal transport seeks the most efficient way to transform one probability distribution into another, typically under constraints that preserve total mass. However, in many practical applications, such as point cloud and image co-registration, the source and target data distributions may have unequal mass. Herein, this is overcome by relaxing the aformentioned constraints. This proposed ``partial'' optimal transport framework adaptively selects and matches subsets of both source and target distributions, thereby enabling robust outlier rejection and noise reduction. The method relaxes the strict constraints typically used in linear programming formulations of optimal transport, thus allowing flexibility on both sides of the matching problem. The resulting transport plan is then refined through branch-and-cut and mass bounding procedures that enforce binary mass assignments and further prune undesired points. For scalability, traditional linear programming solvers are replaced by an efficient gradient-based algorithm. This approach is validated on synthetic two-dimensional examples and real-world three-dimensional point cloud data.