A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration

  • Authors:

  • Xiaoqun Zhang;Martin Burger;Stanley Osher

  • Affiliations:

  • Department of Mathematics, Shanghai Jiao Tong University, Shanghai, P.R. China 200240;Institute for Computational and Applied Mathematics, Westfälische Wilhelms-Universität, Münster, Germany 48163;Department of Mathematics, UCLA, Los Angeles, USA 90095-1555

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2011

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Abstract

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 驴 1 basis pursuit, TV驴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.