Volume 13, Issue 1
Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications

Jianchao Bai, Jicheng Li & Zhie Wu

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 176-199.

Published online: 2019-12

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  • Abstract

By reviewing the primal-dual hybrid gradient algorithm (PDHG) proposed by He, You and Yuan (SIAM J. Image Sci., 7(4) (2014), pp. 2526-2537), in this paper we introduce four improved schemes for solving a class of saddle-point problems. Convergence properties of the proposed algorithms are ensured based on weak assumptions, where none of the objective functions are assumed to be strongly convex but the step-sizes in the primal-dual updates are more flexible than the previous. By making use of variational analysis, the global convergence and sublinear convergence rate in the ergodic/nonergodic sense are established, and the  numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.

  • Keywords

Saddle-point problem, primal-dual hybrid gradient algorithm, variational inequality, convergence complexity, image deblurring.

  • AMS Subject Headings

65K10, 65Y20, 90C90

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

bjc1987@163.com (Jianchao Bai)

jcli@mail.xjtu.edu.cn (Jicheng Li)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-176, author = {Bai , Jianchao and Li , Jicheng and Wu , Zhie }, title = {Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {176--199}, abstract = {

By reviewing the primal-dual hybrid gradient algorithm (PDHG) proposed by He, You and Yuan (SIAM J. Image Sci., 7(4) (2014), pp. 2526-2537), in this paper we introduce four improved schemes for solving a class of saddle-point problems. Convergence properties of the proposed algorithms are ensured based on weak assumptions, where none of the objective functions are assumed to be strongly convex but the step-sizes in the primal-dual updates are more flexible than the previous. By making use of variational analysis, the global convergence and sublinear convergence rate in the ergodic/nonergodic sense are established, and the  numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0030}, url = {http://global-sci.org/intro/article_detail/nmtma/13436.html} }
TY - JOUR T1 - Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications AU - Bai , Jianchao AU - Li , Jicheng AU - Wu , Zhie JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 176 EP - 199 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0030 UR - https://global-sci.org/intro/article_detail/nmtma/13436.html KW - Saddle-point problem, primal-dual hybrid gradient algorithm, variational inequality, convergence complexity, image deblurring. AB -

By reviewing the primal-dual hybrid gradient algorithm (PDHG) proposed by He, You and Yuan (SIAM J. Image Sci., 7(4) (2014), pp. 2526-2537), in this paper we introduce four improved schemes for solving a class of saddle-point problems. Convergence properties of the proposed algorithms are ensured based on weak assumptions, where none of the objective functions are assumed to be strongly convex but the step-sizes in the primal-dual updates are more flexible than the previous. By making use of variational analysis, the global convergence and sublinear convergence rate in the ergodic/nonergodic sense are established, and the  numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.

Jianchao Bai, Jicheng Li & Zhie Wu. (2019). Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 176-199. doi:10.4208/nmtma.OA-2019-0030
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