By fixing the number of within group distances, however, a linear objective function is achieved and the problem becomes of the more common set partitioning kind. With the further assumption of collinear objects, yielding an important subclass of problem, the SPP matrix is unimodular and the simplex method suffices to solve the integer programming problem. A Lagrange relaxation on the constraint for the number of within group distances allows, then, the nonlinear clustering problem to be solved exactly - up to the occurrence of gaps - as a sequence of LP's.

An implementation for small problems using the MPSX of IBM is presented, as are the results from various examples. For the non-collinear case the mapping of data into a single dimension is discussed as are strategies for solving larger problems. The behavior of the Lagrangian - generated function values is examined and found to be exploitable. Useful approximations in the event of gaps are found to be easily computed.