I am reading a paper in which authors said quadratic-constrained optimization is used for the following problem:
\begin{align} &\underset{\mathbf{P},\,\,\epsilon}{\min}\:\epsilon\\ &\mathrm{s.t.}\qquad \Big|||\mathbf{P}-\mathbf{P}_i||_2-d_i\Big|\leq\epsilon\:\: \text{for}\:\:i=1,2,3 \end{align} where $\mathbf{P}$ and $\mathbf{P}_i$ are in $\mathbb{R}^3$ and $d_i$ is scalar for $i=1,2,3$.
I actually don't know how to convert these constraints into quadratic form or reformulate the problem in an appropriate way to use a solver. I really appreciate your help.
Assuming $d_i \ge \epsilon$, rewrite $$|||\mathbf{P}-\mathbf{P}_i||_2-d_i|\leq\epsilon$$ as $$d_i-\epsilon \le ||\mathbf{P}-\mathbf{P}_i||_2 \le d_i+\epsilon$$ and then square all three parts to obtain two quadratic inequality constraints.