A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration

• Published in: Journal of scientific computing Vol. 46; no. 1; pp. 20 - 46
• Main Authors: Zhang, Xiaoqun, Burger, Martin, Osher, Stanley
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.01.2011; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Convergence; Image processing; Iterative methods; Joints; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Optimization; Rice; Theoretical; Inexact Uzawa methods; Bregman iteration; minimization; Proximal point iteration; Saddle point
• ISSN: 0885-7474; 1573-7691; 1573-7691
• DOI: 10.1007/s10915-010-9408-8

In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005 ) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010 ; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009 , CAM Report 09-28, UCLA, March 2009 ; Yin, CAAM Report, Rice University, 2009 ) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009 ). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ 1 basis pursuit, TV− L 2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.