Primal-Dual Decomposition by Operator Splitting and Applications to Image Deblurring

Abstract

We present primal-dual decomposition algorithms for convex optimization problems with cost functions $f(x)+g(Ax)$, where $f$ and $g$ have inexpensive proximal operators and $A$ can be decomposed as a sum of two structured matrices. The methods are based on the Douglas--Rachford splitting algorithm applied to various splittings of the primal-dual optimality conditions. We discuss applications to image deblurring problems with nonquadratic data fidelity terms, different types of convex regularization, and simple convex constraints. In these applications, the primal-dual splitting approach allows us to handle general boundary conditions for the blurring operator. Numerical results indicate that the primal-dual splitting methods compare favorably with the alternating direction method of multipliers, the Douglas--Rachford algorithm applied to a reformulated primal problem, and the Chambolle--Pock primal-dual algorithm.