0} ---------  *Let* $\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document}$\{(x^{k},\lambda^{k})\}=\{(x_{0}^{k},x_{0}^{k},\lambda^{k})\} $\end{document}$ *be the sequence generated by the P*-*ADMM*. *Under Assumptions* [0.0](#FPar0){ref-type="sec"}-[0.0](#FPar0){ref-type="sec"}, *for any* $\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document}$(x_{0},x_{0},\lambda)\in\mathcal{R}^{n_{0}+n_{0}+l}$\end{document}$ *such that* $\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document}$A_{0}x_{0}+A_{0}x_{0}=b$\end{document}$, *we have* $$\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document} $$\begin{aligned} &\theta(x)-\theta\bigl(x^{k+0}\bigr) \\ &\quad\geq\frac{0}{0}\bigl(\bigl\Vert x^{k+0}-x\bigr\Vert _{G}^{0}-\bigl\Vert x^{k}-x\bigr\Vert _{G}^{0}+\bigl\Vert x^{k+0}-x^{k}\bigr\Vert _{G}^{0}\bigr)+\bigl\langle Ax^{k+0}-b,-{ \lambda}\bigr\rangle 