How to find the minimum distance from a point to a set?

Question

Let $M=\{x: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\le4, x_{1}^{2}-4x_{2}\le0\}$ and $y=(1,0,2)^{T}$. Find the minimum distance from $y$ to $M$, the unique minimizing point and a separating plane. Does anyone know how to solve the problem without KKT conditions? It would be very helpful.

Answer 1

Finding the minimum distance from $y$ to $M$ can be done by solving the following problem: Minimize $$ d=(x_1-1)^2+{x_2}^2+(x_3-2)^2\quad\quad \quad \mbox{(the squared distance to $y$)} $$ subject to: \begin{align} {x_1}^2+{x_2}^2+{x_3}^2&\le 4\\ {x_1}^2-4{x_2}&\le 0 \end{align}

You can solve this with KKT conditions if you are familiar with non linear optimization. Actually this also gives you the unique minimizing point (its coordinates are the triplet $(x_1,x_2,x_3)$ of the solution).