  Multi-block separable convex problems recently received considerable
attention. This class of optimization problems minimizes a separable convex
objective function with linear constraints. The algorithmic challenges come
from the fact that the classic alternating direction method of multipliers
(ADMM) for the problem is not necessarily convergent. However, it is observed
that ADMM outperforms numerically many of its variants with guaranteed
theoretical convergence. The goal of this paper is to develop convergent and
computationally efficient algorithms for solving multi-block separable convex
problems. We first characterize the solutions of the optimization problems by
proximity operators of the convex functions involved in their objective
function. We then design a two-step fixed-point iterative scheme for solving
these problems based on the characterization. We further prove convergence of
the iterative scheme and show that it has O(1/k) convergence rate in the
ergodic sense and the sense of the partial primal-dual gap, where k denotes the
iteration number. Moreover, we derive specific two-step fixed-point proximity
algorithms (2SFPPA) from the proposed iterative scheme and establish their
global convergence. Numerical experiments for solving the sparse MRI problem
demonstrate the numerical efficiency of the proposed 2SFPPA.
