Will optimal point of a convex function $f(x)$ under linear constraint lie on boundary?

Question

I have a function $f(x)$ that is positive definite quadratic function. I have linear constraints , then will the optimal lie on boundary ?

My answer that I feel is "No" it will not lay at the boundary. But I am unable to give a solid proof due to my weak calculus. (specially when it comes to higher dimensions)

Can anyone help me visualize from $1$ dimension (taking $f(x)=x^2$) and move towards higher dimension ?

Answer 1

It might lie on the boundary though it need not.

Let's consider a one dimensional example.

$\min x^2$ subject to $x \ge 0$.

The optimal solution is clearly $x=0$ which is on the boundary.

$\min x^2$ subject to $x \ge -1$.

Then the optimal solution is not on the boundary.

In general, given $x^TAx+b^Tx$, differentiating it gives us $2Ax+b$, we can check if the stationary point is in the interior. Suppose not, then the optimal solution must be on the boundary.