ABSTRACT
The theory of alternating optimization presented in this chapter is quite general and pretty technical. This chapter presents two theorems - one for local convergence of AO and one for global convergence of AO - that prescribe the conditions under which we can expect convergence of a very general AO scheme. It is useful and efficient to cover the general theory in some detail for three reasons. The theory in this chapter applies to two of the four clustering algorithms that have been discussed - viz., FCM and GMD/EM. The problems for which AO is a worthy candidate are those for which the simultaneous optimization over all variables is much more difficult than the restricted optimizations over subsets of variables. The field of numerical optimization has changed over the years to reflect changes in the theory, and the evolution of the software and hardware available for implementation has greatly altered this branch of mathematics in the last 40 years.
