A Primal Douglas–Rachford Splitting Method for the Constrained Minimization Problem in Compressive Sensing

Abstract

Compressive sensing has achieved great success in many scientific research fields. It has revealed that sparse signals can be stably recovered from a small number of noisy measurements by solving the constrained convex $${\ell }_1$$ -minimization problem. In practice, a faster algorithm for solving this optimization problem is the key to compressive sensing. The Douglas–Rachford splitting method is a well-known operator splitting method that has been widely applied for solving a certain class of convex composite problems. In particular, its dual application results in the popular alternating direction method of multipliers (ADMM). In this paper, we reformulate the constrained convex $${\ell }_1$$ -minimization problem as a convex composite problem with a special structure and then apply the primal Douglas–Rachford splitting method to solve it. The computational cost of the developed algorithm in each iteration is dominated by the projection onto the constraint set. A fast and efficient method of computing the projection is proposed. Numerical results show that the developed algorithm performs better than the popular NESTA and LADMM (inexact ADMM) in terms of accuracy and run time for large-scale sparse signal recovery.