It is a low-rank sparse decomposition problem: $ D= ML + S + \epsilon$, that we know the matrix D can be decompose into 2 part that one part $L$ is low rank, the other part $S$ is sparse, $\epsilon$ is noise.
I am right now solve this problem with this eqaution with Douglas-Rachford method: $$ \min_{S,L} \quad \lambda_{1}\left\| S \right\|_{1} + \lambda_{2}\left\| L \right\|_{*} + \left\| D-MS-L \right\|_{2}$$
Where $M \in \mathbb{R}^{n \times m}$, $S \in \mathbb{R}^{m \times z}$, $L \in \mathbb{R}^{n \times z}$ and $D \in \mathbb{R}^{n \times z}$
let $$ f(S,L) = \lambda_{1}\left\| S \right\|_{1} + \lambda_{2}\left\| L \right\|_{*} $$
$$g(S,L) = \left\| D-MS-L \right\|_{2}$$
Evaluating the prox-operator of $f$ reduces to separately evaluating the prox-operators of the $L_{1}$ norm and the nuclear norm. for $g(S,L)$:
$$ \text{prox}_{g}(S) = \min _{ x } { \left\| D-MS-L \right\| }_{ 2 }^{ 2 }+\frac { 1 }{ 2\lambda } \left\| S-x \right\| _{ 2 }^{ 2 }$$
$$ \text{prox}_{g}(L) = \min _{ x } { \left\| D-MS-L \right\| }_{ 2 }^{ 2 }+\frac { 1 }{ 2\lambda } \left\| L-x \right\| _{ 2 }^{ 2 }$$
I want to add another constraint that is for every element in $L$ satisfy the constraint $D-L>0$.