We consider the task of optimizing a quadratic objective over binary variables with additional side constraints. We introduce some of these tasks as they appear in different areas, together with general and problem-specific exact solution approaches. The corresponding unconstrained binary quadratic problem is well-known to be equivalent to the maximum cut problem. For some applications, a face of a cut polytope is induced which makes a graph-partitioning-based solution approach feasible. Such approaches are enriched by problem-specific polyhedral knowledge. We focus on exact solution approaches, especially for the quadratic linear ordering problem and the quadratic matching problem which generalizes the quadratic assignment problem.
