An arbitrary starting variable dimension algorithm is proposed to compute an integer point of an n-dimensional simplex. It is based on an integer labeling rule and a triangulation of Rn. The algorithm consists of two interchanging phases. The first phase of the algorithm is a variable dimension algorithm, which generates simplices of varying dimensions, and the second phase of the algorithm forms a full-dimensional pivoting procedure, which generates n-dimensional simplices. The algorithm varies from one phase to the other. When the matrix defining the simplex is in the so-called canonical form, starting at an arbitrary integer point, the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists.
