Abstract We propose a novel discrete solver for optimizing functional map-based energies, including descriptor preservation and promoting structural properties such as area-preservation, bijectivity and Laplacian commutativity among others. Unlike the commonly-used continuous optimization methods, our approach enforces the functional map to be associated with a pointwise correspondence as a hard constraint, which provides a stronger link between optimized properties of functional and point-to-point maps. Under this hard constraint, our solver obtains functional maps with lower energy values compared to the standard continuous strategies. Perhaps more importantly, the recovered pointwise maps from our discrete solver preserve the optimized for functional properties and are thus of higher overall quality. We demonstrate the advantages of our discrete solver on a range of energies and shape categories, compared to existing techniques for promoting pointwise maps within the functional map framework. Finally, with this solver in hand, we introduce a novel Effective Functional Map Refinement (EFMR) method which achieves the state-of-the-art accuracy on the SHREC'19 benchmark.