Multi-way alternating minimization

In a K-way minimization problem, we are interested in finding min (x₁∈S₁) ··· min (x_K ∈S_K) f(x₁, ···, x _K), where f is continuous and bounded from below, and S_i is a compact convex set in IR(n_i), 1⩽i⩽K. In a paper by Csiszar and Tusnady (1984), a similar problem with somewhat less stringent conditions was studied for K=2, where it was shown that an alternating minimization algorithm converges to the infimum provided a certain geometric condition is satisfied. In this paper, we take an approach (also with strong geometric flavor) different from theirs, which enables us to obtain a sufficient condition for an alternating minimization algorithm to converge to the minima. In particular, we show that it is sufficient for f to be convex. The Arimoto-Blahut algorithm for computing channel capacity is discussed as an example of application of our results