A linearly convergent proximal ADMM with new iterative format for BPDN in compressed sensing problem

In recent years, compressive sensing (CS) problem is being popularly applied in the fields of signal processing and statistical inference. The alternating direction method of multipliers (ADMM) is applicable to the equivalent forms of basis pursuit denoising (BPDN) in CS problem. However, the solving speed and accuracy are adversely affected when the dimension increases greatly. In this paper, a new iterative format of proximal ADMM, which has fast solving speed and pinpoint accuracy when the dimension increases, is proposed to solve BPDN problem. Global convergence of the new type proximal ADMM is established in detail, and we exhibit a $R-$ linear convergence rate under suitable condition. Moreover, we apply this new algorithm to solve different types of BPDN problems. Compared with the state-of-the-art of algorithms in BPDN problem, the proposed algorithm is more accurate and efficient.