Title: A general framework for solving convex optimization problems involving the sum of three convex functions,
Abstract: 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. This type of optimization problem has wide application
in signal recovery and image processing. To make full use of the
differentiability function in the optimization problem, we take advantage of
two operator splitting methods: the forward-backward splitting method and the
three operator splitting method. In the iteration scheme derived from the two
operator splitting methods, we need to compute the proximity operator of $g+h
\circ B$ and $h \circ B$, respectively. Although these proximity operators do
not have a closed-form solution in general, they can be solved very
efficiently. We mainly employ two different approaches to solve these proximity
operators: one is dual and the other is primal-dual. Following this way, we
fortunately find that three existing iterative algorithms including Condat and
Vu algorithm, primal-dual fixed point (PDFP) algorithm and primal-dual three
operator (PD3O) algorithm are a special case of our proposed iterative
algorithms. Moreover, we discover a new kind of iterative algorithm to solve
the considered optimization problem, which is not covered by the existing ones.
Under mild conditions, we prove the convergence of the proposed iterative
algorithms. Numerical experiments applied on fused Lasso problem, constrained
total variation regularization in computed tomography (CT) image reconstruction
and low-rank total variation image super-resolution problem demonstrate the
effectiveness and efficiency of the proposed iterative algorithms.