  The forward-backward operator splitting algorithm is one of the most
important methods for solving the optimization problem of the sum of two convex
functions, where one is differentiable with a Lipschitz continuous gradient and
the other is possibly nonsmooth but proximable. It is convenient to solve some
optimization problems in the form of dual or primal-dual problems. Both methods
are mature in theory. In this paper, we construct several efficient first-order
splitting algorithms for solving a multi-block composite convex optimization
problem. The objective function includes a smooth function with a Lipschitz
continuous gradient, a proximable convex function that may be nonsmooth, and a
finite sum of a composition of a proximable function and a bounded linear
operator. To solve such an optimization problem, we transform it into the sum
of three convex functions by defining an appropriate inner product space. On
the basis of the dual forward-backward splitting algorithm and the primal-dual
forward-backward splitting algorithm, we develop several iterative algorithms
that involve only computing the gradient of the differentiable function and
proximity operators of related convex functions. These iterative algorithms are
matrix-inversion-free and completely splitting algorithms. Finally, we employ
the proposed iterative algorithms to solve a regularized general prior image
constrained compressed sensing (PICCS) model that is derived from computed
tomography (CT) image reconstruction under sparse sampling of projection
measurements. Numerical results show that the proposed iterative algorithms
outperform other algorithms.
