An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions

In recent years, the optimization problem of the sum of several convex functions has received much attention. In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a closed-form expression of their proximity operators and $B$ is a bounded linear operator. Such optimization problems have wide application in signal recovery and image processing. To make full use of the differentiable function in the problem, we propose several inner-outer iterative algorithms based on the forward-backward splitting algorithm and the three-operator splitting algorithm frameworks. In the process of deriving the iterative algorithms, we use dual and primal-dual methods to solve the proximity operator of the functions $~g~+~h~\circ~B~$ and $~h~\circ~B~$. Under mild assumptions on the parameters, we prove the convergence of the proposed iterative algorithms. By comparing with the Condat and Vu algorithm, the primal-dual fixed point (PDFP) algorithm and the primal-dual three-operator (PD3O) algorithm, we establish the connection between these algorithms with ours. Numerical experiments applied to the fused Lasso problem, the constrained total variation regularization problem and the low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms.