Issue No. 01 - January (1984 vol. 6)
M. A. Ismail , Department of Computer Science, University of Windsor, Windsor, Canada.
Shokri Z. Selim , Department of Systems Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia.
The K-means algorithm is a commonly used technique in cluster analysis. In this paper, several questions about the algorithm are addressed. The clustering problem is first cast as a nonconvex mathematical program. Then, a rigorous proof of the finite convergence of the K-means-type algorithm is given for any metric. It is shown that under certain conditions the algorithm may fail to converge to a local minimum, and that it converges under differentiability conditions to a Kuhn-Tucker point. Finally, a method for obtaining a local-minimum solution is given.
M. A. Ismail, Shokri Z. Selim, "K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 6, no. , pp. 81-87, January 1984, doi:10.1109/TPAMI.1984.4767478