As @ManfredWeis noted, the objective is equivalent to minimizing $\sum_{i=1}^K \frac{x_i^2}{y_i}$.  A natural reformulation introduces binary decision variables $u_{ij}$ for $i\in\{1,\dots,K\}$ and $j\in \{0,\dots, X\}$ and $v_{ik}$ for $i\in\{1,\dots,K\}$ and $k\in \{1,\dots, Y\}$.  The problem is to minimize the quadratic function
$$\sum_{i=1}^K \sum_{j=0}^X \sum_{k=1}^Y \frac{j^2}{k} u_{ij} v_{ik} \tag1\label1$$
subject to linear constraints
\begin{align}
\sum_i x_i &= X \\
\sum_i y_i &= Y \\
x_i &\le y_i &&\text{for all $i$} \\
\sum_j u_{ij} &= 1 &&\text{for all $i$} \\
\sum_j j u_{ij} &= x_i &&\text{for all $i$} \\
\sum_k v_{ik} &= 1 &&\text{for all $i$} \\
\sum_k k v_{ik} &= y_i &&\text{for all $i$}
\end{align}
You can call an MIQP (or BQP) solver.

Alternatively, you can linearize \eqref{1} by introducing nonnegative variables $w_{ijk}$ to represent the product $u_{ij} v_{ik}$.
You can then minimize $$\sum_{i=1}^K \sum_{j=0}^X \sum_{k=1}^Y \frac{j^2}{k} w_{ijk} \tag{1p}\label{1'}$$
The usual linearization imposes linear constraints
\begin{align}
w_{ijk} &\ge u_{ij} + v_{ik} - 1 \tag2\label2 \\
w_{ijk} &\le u_{ij} \tag3\label3 \\
w_{ijk} &\le v_{ik} \tag4\label4
\end{align}
But you can omit \eqref{3} and \eqref{4} because the objective will drive these constraints to be satisfied naturally.

A more compact linearization replaces \eqref{2} with
\begin{align}
\sum_j w_{ijk} &= v_{ik} &&\text{for all $i$ and $k$} \\
\sum_k w_{ijk} &= u_{ij} &&\text{for all $i$ and $j$}
\end{align}