I have a problem on hand that I am not sure how to handle it. Can you please help/guide me to solve this optimization problem?
\begin{align} \arg\min_{x \in \mathbb{R}^n} I_C(x) + \alpha \sum_i \left\|y_i -x \right\|_2^2 \ , \end{align} where $I_C(x)$ is an indicator or characteristic function and the set $C$ can be a norm-2 ball, i.e., $C = \left\{x \in \mathbb{R}^n : \left\|x - c \right\|_2^2 \leq r\right\}$, $y_i, c \in \mathbb{R}^n$ are given, and $r \in \mathbb{R}$.
My partial attempt:
The above problem can be rewritten as \begin{equation} \label{eqn:weighted_projection_problem_definition_3} \begin{aligned} & \underset{x}{\text{minimize}} & & \alpha \sum_i \left\|y_i -x \right\|_2^2 \\ & \text{subject to} & & \left\|x - c \right\|_2^2 \leq r \ ,\\ %&&& X \succeq 0. \end{aligned} \end{equation}
Right?
If yes, then I think it is possible to solve in the closed form.