Discrete Morse theory was developed by Forman as a combinatorial analog to the classical smooth Morse theory. There are numerous applications to questions in combinatorial topology and related fields. It turns out that the topologically relevant information of a discrete Morse function on a simplicial complex can be encoded as a (partial) matching in its Hasse diagram, the corresponding Morse matching. A matching in the Hasse diagram is Morse if it satisfies a certain, entirely combinatorial, acyclicity condition. The goal of this project is to compute Morse matchings of maximal cardinality via combinatorial optimization methods.