Minimize $\|\mathbf{x-y}\|^2 $ subject to $x \in $ set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$

Question

We are given the set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$ and a point $\mathbf{y} \in \mathbb{R}^n$.

Our goal is to find point $\mathbf{\hat{x}}$ which minimizes $\|\mathbf{x-y}\|_2^2$.

• If $\|\mathbf{x-y}\|_2^2\leq r^2$ then point $\mathbf{y}$ lies in set and the distance is $0$
• If $\|\mathbf{x-y}\|_2^2 > r^2$ then point $\mathbf{y}$ lies somewhere outside of $S$. That means the the point in question $\mathbf{\hat{x}}$ must be on the set S , so it follows that $\|\mathbf{\hat{x}-x_c}\|^2=r^2$

Finally, the problem can be formulated as \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x)=\|\mathbf{x-y}\|^2 \\ & \text{subject to} & & h(x) = \|\mathbf{x-x_c}\|^2-r^2=0 \end{aligned} \end{equation*}

What optimization method can I use to solve the above problem? I have solved problems with linear equlity constraints but this problem is quadratically constrained.

Is there any way to transform the problem to be linearly constrained?