since $\bmT_{\gamma}$ is an elementwise operator, we can update each column of $\overline{\bsH}_{n+1}\bC_{n+1}^T+\left(\beta_3/\mu\right)\bsD_n$ in \[Wsubexplicit\] directly after updating each row of $\bC_{n+1}$ by \[csubexplicit\].

Convergence analysis {#ConvergenceAnalysis}
--------------------

In this subsection, we focus on the convergence analysis of the sequence $\left(\bsv_n,\bc_n,\bsW_n\right)$ generated by \[Alg1\] based on the frameworks in [@H.Attouch2010; @C.Bao2016; @J.Bolte2014; @Kurdyk1998; @Lojasiewicz1995; @Y.Xu2013] for the nonconvex and nondifferentiable optimization. To do this, we begin with introducing some basic notation and definitions.

Let $f:\R^n\to\R\cup\{\infty\}$ be a proper and lower semicontinuous (lsc) function.

**

1.  The domain of $f$, denoted $\dom(f)$, is defined as $$\begin{aligned}
    \dom(f)=\{\bu\in\R^n:f(\bu)<\infty\}.\end{aligned}$$

2.  For each $\bu\in\dom(f)$, the Fréchet subdifferential of $f$ at $\bu$ is defined as $$\begin{aligned}
    \p_Ff(\bu)=\left\{\bs\in\R^n:\liminf_{\bsv\to\bu}\f{f(\bsv)-f(\bu)-\la\bs,\bsv-\bu\ra}{\|\bsv-\bu\|}\geq0\right\}.\end{aligned}$$ If $\bu\notin\dom(f)$, then we set $\p_Ff(\bu)=\emptyset$.

3.  The (limiting-) subdifferential of $f$ at $\bu$ is defined as $$\begin{aligned}
    \p f(\bu)=\left\{\bs\in\R^n:\exists \bu_n~\text{s.t.}~f(\bu_n)\to f(\bu)~\&~\bs_n\in\p_Ff(\bu_n)\to\bs\right\}.\end{aligned}$$ The domain of $\p f$ is defined as $\dom(\p f)=\left\{\bu\in\R^n:\p f(\bu)\neq\emptyset\right\}$.

4.  $\bu\in\dom(f)$ is a critical point of $f$ if $\0\in\p f(\bu)$.

\[RK3\] Since the minimization of a real-valued complex variable objective function in \[ProposedCSMRIModel\] is based on the Wirtinger calculus (i.e. by identifying $\C\simeq\R^2$), the theoretical results on the proximal schemes for the optimization on the real vector space are still applicable.

The global convergence 