An algorithm for the verification of composite position tolerances is proposed and its computer implementation is described. It is shown that a prevalent graphical verification method does not adequately reflect the standard. Composite position tolerance requirements are mathematically formulated as a non-linear optimization problem with a linear objective function and quadratic constraints. The method of feasible directions is used as the basis for an efficient computer implementation. Its performance is evaluated under various test conditions as well as against a simplex pattern search.