Journal of Mathematical Imaging and Vision

, Volume 58, Issue 2, pp 279–293

An Algorithmic Framework of Generalized Primal–Dual Hybrid Gradient Methods for Saddle Point Problems

Article

DOI: 10.1007/s10851-017-0709-5

Cite this article as:
He, B., Ma, F. & Yuan, X. J Math Imaging Vis (2017) 58: 279. doi:10.1007/s10851-017-0709-5

Abstract

The primal–dual hybrid gradient method (PDHG) originates from the Arrow–Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for the special case where the combination parameter is 1, the PDHG cannot be casted to the proximal point algorithm framework due to the lack of symmetry in the matrix associated with the proximal regularization terms. The PDHG scheme is nonsymmetric also in the sense that one variable is updated twice while the other is only updated once at each iteration. These nonsymmetry features also explain why more theoretical issues remain challenging for generalized PDHG schemes; for example, the worst-case convergence rate of PDHG measured by the iteration complexity in a nonergodic sense is still missing. In this paper, we further consider how to generalize the PDHG and propose an algorithmic framework of generalized PDHG schemes for saddle point problems. This algorithmic framework allows the output of the PDHG subroutine to be further updated by correction steps with constant step sizes. We investigate the restriction onto these step sizes and conduct the convergence analysis for the algorithmic framework. The algorithmic framework turns out to include some existing PDHG schemes as special cases, and it immediately yields a class of new generalized PDHG schemes by choosing different step sizes for the correction steps. In particular, a completely symmetric PDHG scheme with the golden-ratio step sizes is included. Theoretically, an advantage of the algorithmic framework is that the worst-case convergence rate measured by the iteration complexity in both the ergodic and nonergodic senses can be established.

Keywords

Convex programming Saddle point problem Image restoration Variational models Primal–dual hybrid gradient method Convergence rate Variational inequalities 

Funding information

Funder NameGrant NumberFunding Note
National Natural Science Foundation of China
  • 11471156
  • 91530115
National Natural Science Foundation of China
  • 71401176
Research Grants Council, University Grants Committee (HK)
  • HKBU12300515

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Department of MathematicsSouth University of Science and Technology of ChinaShenzhenChina
  3. 3.High-Tech Institute of Xi’anXi’anChina
  4. 4.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong

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