Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications

Authors

  • Jianchao Bai Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, Shanxi, China
  • Jicheng Li School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China
  • Zhie Wu School of Electronics and Information, Xi’an Polytechnic University, Xi’an 710048, Shaanxi, China

DOI:

https://doi.org/10.4208/nmtma.OA-2019-0030

Keywords:

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

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.

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Published

2020-03-06

Issue

Vol. 13 No. 1 (2020)

Section

Articles