There is a plethora of literature in proximal operators and proximal optimization algorithms specially for Compressive sensing. A proximal operator is defined as \begin{equation} \operatorname{prox}_f(x) = \arg\min_z \left(f(x)+\frac{1}{2}\left\|x-z\right\|^2\right), \end{equation} where $x\in\mathbb{R}^n$.
Are proximal operators and algorithms also used for optimizing complex real-valued functions? In my mind any real-valued function whose domain is in the world of complex numbers can always be transformed to one with real numbers. However, surprisingly, the literature seems to ignore the world of complex numbers.
In particular I am interested in solving the convex optimization problem \begin{align} \text{minimize} &\, \|\mathbf{X}\|_*\\ \text{subject to} &\ \sum_{l=1}^L\left\|b_l-\mathbf{e}_l^H\mathbf{x}_l\right\|^2, \end{align} where $\mathbf{X}=[\mathbf{x}_1,\ldots,\mathbf{x}_L]\in\mathbb{C}^{N\times L}$, $\mathbf{e}_l$ are vectors whose entries have magnitude one, and $(\cdot)^H$ is the conjugate transpose operators. I wanted to use the Douglas-Rachford splitting algorithm (see (38) in page 9 of Combettes and Pesquet's paper) by writing the previous problem as \begin{equation} \text{minimize} \, \|\mathbf{X}\|_* +\operatorname{i}_{\beta}\left(\mathbf{X}\right), \end{equation} where $\operatorname{i}$ is the indicator function and $\beta$ is the convex set defined by the previous constraint. Such algorithm requires the proximal operator of the nuclear norm which can be found in this paper by Candès while the proximal operator of the indicator function is just the projection.
However, as a non-mathematician, I am unsure if I can really employ this methods when working with complex numbers. Do the proximal operators change? Or they have the same form than in the case of real variables?
