We present several mathematical-optimization formulations for a problem that commonly occurs in geometry processing and specifically in the design of so-called smooth direction fields on surfaces. This problem has direct applications in 3D shape parameterization, texture mapping, and shape design via rough concept sketches, among many others. A key challenge in this setting is to design a set of unit-norm directions, on a given surface, that satisfy some prescribed constraints and vary smoothly. This naturally leads to mixed-integer optimization formulations, because the smoothness needs to be formulated with respect to angle-valued variables, which to compare one needs to fix the discrete jump between nearby points. Previous works have primarily attacked this problem via a greedy ad-hoc strategy with a specialized solver. We demonstrate how the problem can be cast in a standard mathematical-optimization form, and we suggest several relaxations that are especially adapted to modern mathematical-optimization solvers.