Many hard discrete optimization problems can be handled with doubly nonnegative, DNN, relaxations, i.e.,~positive semidefinite and nonnegative elementwise relaxations. The size and complexity of these relaxations grows quickly. However, one can exploit the special structure of the relaxations and take advantage of natural splittings. We look at many special classes of problems. This includes: discrete optimization problems such as quadratic assignment and graph partitioning and clustering; and distance geometry problems such as molecular conformation; general barycenter problems.