Analyzing the associated minimization problem of an augmented system I am interested in the following problem:
$$\min \|u\|_2^2 + \|v\|_2^2 \text{ such that } Ru + Sv = c.$$
Here $R \in \mathbb{R}^{r \times r}$ is upper triangular and invertible, $S \in \mathbb{R}^{r \times (n-r)}$ is just a matrix with no special properties, $c \in \mathbb{R}^r$, $u \in \mathbb{R}^r, v \in \mathbb{R}^{n-r}.$
I am unsure of the best way to approach this problem. If you write $u = R^{-1}(c-Sv)$, and attempt to minimize
$$\min_v \|R^{-1}(c-Sv)\|_2^2 + \|v\|_2^2$$
you get the first order criticality condition
$$0 = 2v + 2(R^{-1}S)^TR^{-1}Sv - 2(R^{-1}S)^Tc.$$
I don't have any idea how to solve this, so it's probably not useful.
Alternatively we could consider the Lagrangian $$L = \|u\|_2^2 + \|v\|_2^2 + 2\langle \lambda, Ru + Sv - c\rangle.$$
This yields first order conditions
$$Ru + Sv = c$$
$$u + R^T\lambda = 0$$
$$v + S^T\lambda =0.$$
Then we could try to solve for $\lambda:$
$$-(RR^T + SS^T)\lambda = c.$$
Since $R$ is invertible, $RR^T + SS^T$ is positive definite, so it's invertible too. But defining $\lambda = (RR^T + SS^T)^{-1}(-c)$ is not so helpful.
I'm not looking for a full solution, just a tip on how to proceed. Thanks!
 A: The objective function is convex and the constraints are linear.
Hence this is a convex QP in $u,v$, and strong duality holds.
Defining the Lagrangian
$$
L(u,v;\lambda) := \|u\|_2^2 + \|v\|_2^2 - \lambda^\top(Ru + Sv - c),
$$
we have that the original problem is equivalent to
$$
\min_x\max_\lambda \{L(u,v;\lambda)\}
$$
and that by strong duality
$$
\min_x\max_\lambda \{L(u,v;\lambda)\} = \max_\lambda \min_x L(u,v;\lambda)
$$
(though in general we have "$\geq$" by weak duality, i.e., from the minmax inequality).
We can obtain the dual problem in $\lambda$ by plugging in $u = R^\top \lambda$ and $v=S^\top\lambda$ (from stationarity conditions) into the Lagrangian (effectively minimizing the Lagrangian wrt $u,v$) to obtain the dual
\begin{align}
\max_\lambda\; -\tfrac12 \|R^\top \lambda\|_2^2-\tfrac12 \|S^\top \lambda\|_2^2 - \lambda^\top c
\end{align}
Once we find $\lambda^*$, then we can recover optimal $u,v$ from $u^* = R^\top \lambda^*$ and similarly $v^* = S^\top\lambda^*$.

Aside from that approach, what you describe by eliminating $u$ and then solving for $v$ from the linear system works just fine.
