The detection phase in computational contact mechanics can be subdivided into a global search and a local detection. When potential contact is detected by the former, a rigorous local detection determines which surface elements come or may come in contact in the current increment. We first introduce a rigorous definition of the closest point for non-differentiable lower-dimensional manifolds. We then simplify the detection by formulating an optimization problem subject to inequality constraints. The formulation is then solved using different techniques from the field of mathematical optimization, for both linear and quadratic finite element meshes. The resulting general and robust detection scheme is tested on a set of problems and compared with other techniques commonly used in computational geometry.