  The alternating minimization (AM) method is a fundamental method for
minimizing convex functions whose variable consists of two blocks. How to
efficiently solve each subproblems when applying the AM method is the most
concerned task. In this paper, we investigate this task and design two new
variants of the AM method by borrowing proximal linearized techniques. The
first variant is very suitable for the case where half of the subproblems are
hard to be solved and the other half can be directly computed. The second
variant is designed for parallel computation. Both of them are featured by
simplicity at each iteration step. Theoretically, with the help of the proximal
operator we first write the new as well as the existing AM variants into
uniform expressions, and then prove that they enjoy sublinear rates of
convergence under very minimal assumptions.
