ADMM variant with projections at each update I have the following convex problem
$$
\begin{align*}
\underset{x \in \mathcal{X}, y \in \mathcal{Y}}{\text{minimize}} &\quad f(x) + g(y) \\
\text{subject to} &\quad Ax + By = c
\end{align*}
$$
where $\mathcal{X} \subseteq \mathbb{R}^{n}$ and $\mathcal{Y} \subseteq \mathbb{R}^{m}$ are convex sets.
In addition, $f$ and $g$ are strongly convex twice-differentiable functions, also $A$ and $B$ are full rank.
This problem has the augmented Lagrangian, for $\rho > 0$
$$
L_\rho (x, y, z) = f(x) + g(y) + z^\top (Ax + By - c) + \frac{\rho}{2} \lVert Ax + By - c \rVert_2 ^2.
$$
In my particular problem instance, it happens to be easy to calculate both
$$
\begin{align*}
\underset{x \in \mathbb{R}^n}{\text{argmin}} \, L_\rho (x, y, z)  &\text{ when $y \in \mathcal{Y}, z \in \mathbb{R}^p$ are fixed} \\
\underset{y \in \mathbb{R}^m}{\text{argmin}} \, L_\rho (x, y, z) &\text{ when $x \in \mathcal{X}, z \in \mathbb{R}^{p}$ are fixed},
\end{align*}
$$
and it is also simple to implement the projections $\Pi_{\mathcal{X}} : \mathbb{R}^{n} \to \mathcal{X}$ and $\Pi_{\mathcal{Y}} : \mathbb{R}^m \to \mathcal{Y}$.
As such, I would like to use the following variant of ADMM:
$$
\begin{align*}
x^{(k+1)} &= \Pi_{\mathcal{X}} \left[ \underset{x \in \mathbb{R}^n}{\text{argmin}} \, L_\rho (x, y^{(k)}, z^{(k)}) \right] \\
y^{(k+1)} &= \Pi_{\mathcal{Y}} \left[ \underset{y \in \mathbb{R}^m}{\text{argmin}} \, L_\rho (x^{(k+1)}, y, z^{(k)}) \right] \\
z^{(k+1)} &= z^{(k)} + \rho (Ax^{(k+1)} + B y^{(k+1)} - c).
\end{align*}
$$
However, I have had trouble finding (or perhaps understanding?) literature that proves convergence under this scheme.
I also had trouble extending some existing proofs of ADMM convergence to this case.
Is convergence in this minimize-then-project setting known? If not, what are some things I should try when extending the proof of existing techniques? What additional assumptions might be helpful?
 A: HINT:
Reformulate your original problem
\begin{align*}
\underset{x \in \mathcal{X}, y \in \mathcal{Y}}{\text{minimize}} &\quad f(x) + g(y) \\
\text{subject to} &\quad Ax + By = c
\end{align*}
as
\begin{align*}
\underset{x \in \mathbb{R}^n, y \in \mathbb{R}^m}{\text{minimize}} &\quad \underbrace{f(x) + \delta_{\mathcal{X}}(x)}_{:= \overline{f}(x)}  + \underbrace{ g(y) + \delta_{\mathcal{Y}}(y)}_{:= \overline{g}(y)} \\
\text{subject to} &\quad Ax + By = c,
\end{align*}
where $\delta_C(z)$ is an indicator function to the set $C$ such that $\delta_C(z)=0$ when $z \in C$ otherwise $+\infty$.
Thus,
\begin{align*}
\underset{x \in \mathbb{R}^n, y \in \mathbb{R}^m}{\text{minimize}} &\quad \overline{f}(x)  + \overline{g}(y) \\
\text{subject to} &\quad Ax + By = c.
\end{align*}
Now, I think you can try to solve this problem with classical ADMM (that has a composite sum of two functions) as mentioned in Boyd et al.'s monograph since you have nice quadratic functions $f$ and $g$. The convergence proof holds for this case as well.
If you do not manage to solve with the classical ADMM, then you may try to solve it using so-called function linearized proximal ADMM method (referred to as FLiP-ADMM) described in Chapter 8 of this draft book. This FLiP-ADMM solves the problem of form
\begin{align*}
\underset{x \in \mathbb{R}^n, y \in \mathbb{R}^m}{\text{minimize}} &\quad f_1(x) + f_2(x) + g_1(y) + g_2(y) \\
\text{subject to} &\quad Ax + By = c,
\end{align*}
where $f_1$ and $g_1$ are closed convex proper (not necessarily differentiable such as the indicator function $\delta$), and $f_2$ and $g_2$ are $L$-smooth functions.
Disclaimer: I have never implemented FLiP-ADMM myself. So, I am not sure how it works in practice, but looks promising.
