Question: I encounter a nonlinear constraint that contains the square root of a sum of integer variables. Of course one could use nonlinear solvers and techniques; but I like linear programming. Are there any standard results on linearizing or approximating a square root of the sum of integer variables?

For example, the constraints look like this:

$$\sqrt{\sum\_{i \in \mathcal{I}} a\_{ij}x\_{ij} } \leq \theta\_j, \quad \quad \forall j \in \mathcal{J}$$

where $x\_{ij} \in \{0,1\}$ are binary variables, $\theta\_j \in \mathbb{R}$ are continuous variables, and $a\_{ij} \geq 0$ are parameters. $\mathcal{I}$ and $\mathcal{J}$ are any given sets of polynomial size. 

Of course, this constraint is part of a larger MIP, but as I am curious to general methods and results regarding this constraint I believe it not to be of interest to post it here.

Answer: 
This can be handled as an MISOCP, Mixed-Integer Second Order Cone problem. The leading commercial MILP solvers can also handle MISOCP.

Specifically, due to $x\_{ij}$ being binary, $x\_{ij}^2 = x\_{ij}$. Therefore, the left-hand side is the two-norm of the vector over $i \in I$ having elements $\sqrt{a\_{ij}} x\_{ij}$. 

I don't know whether this is the best way to handle this constraint, but it is a way, and it is "exact".
You can manipulate this inequality as follows

$$\sqrt{\sum\_{i \in \mathcal{I}} a\_{ij}x\_{ij} } \leq \theta\_j, \quad \quad \forall j \in \mathcal{J}$$

$$\sum\_{i \in \mathcal{I}} a\_{ij}x\_{ij} \leq \theta\_j^2, \quad \quad \forall j \in \mathcal{J}$$

Now, you need to linearize $\theta\_j^2$ using [McCormick Envelopes](https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes). To do this, assume $-M\_j\leq \theta\_j \leq M\_j$ and consider $w\_j=\theta\_j^2$

$$
\begin{align}
0\leq (\theta\_j + M\_j)(\theta\_j + M\_j) & \implies & -w\_j - 2M\_j\theta \leq M\_j^2\\
0\leq (M\_j - \theta\_j)(M\_j - \theta\_j) & \implies & -w\_j + 2M\_j\theta \leq M\_j^2\\
0\leq (\theta\_j + M\_j)(M\_j - \theta\_j) & \implies & w\_j \leq M\_j^2\\
\end{align}
$$

The final set of constraints is 

$$
\begin{align}
\sum\_{i \in \mathcal{I}} a\_{ij}x\_{ij} \leq w\_j, \quad \quad \forall j \in \mathcal{J}\\
-w\_j - 2M\_j\theta\_j \leq M\_j^2, \quad \quad \forall j \in \mathcal{J}\\
-w\_j + 2M\_j\theta\_j \leq M\_j^2, \quad \quad \forall j \in \mathcal{J}\\
0 \leq w\_j \leq M\_j^2, \quad \quad \forall j \in \mathcal{J}\\
-M\_j \leq \theta\_j \leq M\_j, \quad \quad \forall j \in \mathcal{J}\\
\end{align}
$$

OBS: Verify my counts, please.