@article{639e484b007442bcac6f372e9ad2f24e,

title = "A Peaceman–Rachford Splitting Method with Monotone Plus Skew-Symmetric Splitting for Nonlinear Saddle Point Problems",

abstract = "This paper is devoted to solving the linearly constrained convex optimization problems by Peaceman–Rachford splitting method with monotone plus skew-symmetric splitting on KKT operators. This approach generalizes the Hermitian and skew-Hermitian splitting method, an unconditionally convergent algorithm for non-Hermitian positive definite linear systems, to the nonlinear scenario. The convergence of the proposed algorithm is guaranteed under some mild assumptions, e.g., the strict convexity on objective functions and the consistency on constraints, even though the Lions–Mercier property is not fulfilled. In addition, we explore an inexact version of the proposed algorithm, which allows solving the subproblems approximately with some inexactness criteria. Numerical simulations on an image restoration problem demonstrate the compelling performance of the proposed algorithm.",

keywords = "Contraction, Hermitian and skew-Hermitian splitting method, Image restoration, Inexact method, Parallel computing, Peaceman–Rachford splitting method, Saddle point problem",

author = "Weiyang DING and NG, {Kwok Po} and Wenxing Zhang",

note = "Funding Information: The authors would like to thank the anonymous referees for their valuable comments. W.Y. Ding is partially supported by NSFC Grants 11801479, HKRGC GRF 12301619 and HKBU RC-NACAD-DW. M.K. Ng is partially supported by HKRGC GRF 12306616, 12200317, 12300218 and 12300519. W.X. Zhang is partially supported by NSFC Grant 11571074. 1 The PRSM in [ 21 , 37 ] extended the range of applicability of classical PRSM for positive definite linear system to maximal monotone nonlinear system. In this sense, the recursion ( 1.5 ) with H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} and S \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document} being positive semi-definite or skew-symmetric matrices can also be deemed as PRSM because those matrices are essentially monotone operators. Henceforth, the terminologies “PRSM” and “DRSM” are named after the literature [ 21 , 37 ]. 2 A function θ : R n → ( - ∞ , + ∞ ] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta :\mathbb {R}^n\rightarrow (-\infty ,+\infty ]$$\end{document} is proper if its domain dom ( θ ) : = { x ∈ R n ∣ θ ( x ) < + ∞ } \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {dom}(\theta ):=\{x\in \mathbb {R}^n\mid \theta (x)<+\infty \}$$\end{document} is nonempty, and it is said to be closed if its epigraph epi ( θ ) : = { ( x , y ) ∈ R n × R ∣ θ ( x ) ≤ y } \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {epi}(\theta ):=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}\mid \theta (x)\le y\}$$\end{document} is closed. 3 The μ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} -strongly monotonicity and L -Lipschitz continuity on ∇ θ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$",

year = "2019",

month = nov,

day = "1",

doi = "10.1007/s10915-019-01034-w",

language = "English",

volume = "81",

pages = "763--788",

journal = "Journal of Scientific Computing",

issn = "0885-7474",

publisher = "Springer New York",

number = "2",

}