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My question is similar to this one though a bit more complicated. Though my question also includes indices, I am removing them to ease readability.

Let binary variables $x,y\in\{0,1\}$, non-negative continuous variable $z\in\mathbb{R}^+$ and the parameter $\lambda\in\mathbb{R}^+$. Is there a way to linearize the below equality constraint?

$$\displaystyle z=\sqrt{\lambda\left(x+y\right)}$$

Can we benefit from the fact that $\alpha=x+y$, where $\alpha \in \{0,1,2\}$ and write additional constraints?

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For $j\in\{0,1,2\}$, introduce binary variable $w_j$ to indicate whether $x+y=j$, and then impose the following linear constraints: \begin{align} \sum_{j=0}^2 w_j &= 1 \\ \sum_{j=0}^2 j\cdot w_j &= x+y \\ \sqrt{\lambda}\sum_{j=0}^2 \sqrt{j}\cdot w_j &= z \end{align}

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