An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

  • Zhao DengEmail author
  • Sanyang Liu


The strictly contractive Peaceman–Rachford splitting method (SCPRSM) has received a tremendous amount of attention for solving linearly constrained separable convex optimization problems. In this paper, we propose an indefinite proximal SCPRSM with substitution procedure (abbreviated as PPRSM-S) to improve numerical results. The prediction step takes a proximal SCPRSM cycle to update the variable blocks, then the correction step corrects the output slightly by computing a combination of the prediction step and the previous iteration. We derive the global convergence of the proposed method and analyze the convergence rate results under much mild conditions. Some experimental results on LASSO and total variation-based denoising problems demonstrate the efficiency of the substitution step and the indefinite proximal term.


Convex programming Peaceman–Rachford splitting method Substitution Variational inequality Global convergence 

Mathematics Subject Classification

90C25 90C30 94A08 



This work was supported by the National Natural Science Foundation (no. 61877046) and the Fundamental Research Fund of Xidian University (no. JB180707).

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China

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