  The classical alternating minimization (or projection) algorithm has been
successful in the context of solving optimization problems over two variables.
The iterative nature and simplicity of the algorithm has led to its application
to many areas such as signal processing, information theory, control, and
finance. A general set of sufficient conditions for the convergence and
correctness of the algorithm is quite well-known when the underlying problem
parameters are fixed. In many practical situations, however, the underlying
problem parameters are changing over time, and the use of an adaptive algorithm
is more appropriate. In this paper, we study such an adaptive version of the
alternating minimization algorithm. As a main result of this paper, we provide
a general set of sufficient conditions for the convergence and correctness of
the adaptive algorithm. Perhaps surprisingly, these conditions seem to be the
minimal ones one would expect in such an adaptive setting. We present
applications of our results to adaptive decomposition of mixtures, adaptive
log-optimal portfolio selection, and adaptive filter design.
