I am new to optimization problems but I have this exercise that I cannot solve.
I have the following constrained optimization problem in $\mathbb{R}^2$:
$\min_{(x,y)\in\mathbb{R}^2} x^2+(y-2)^2$
s.t.
$x^2+y^2\leq 1$
I really don't know how to start. A long time ago, in an analysis lecture, we have studied the Lagrange method.
The Problem is that I know how to use it only for a $=$case and not for a $\leq$-case.
Can someone help me?
Extremal points (of a continuously differentiable function) can either be found in the interior of the domain or at the boundary.
To find those in the interior you need to set the gradient to zero:
$$\nabla f =\begin{pmatrix}2x\\2(y-2) \end{pmatrix}$$ and you notice that the extremum is at $(0,2)$ which is outside the considered range.
Now you look for extremal points on the boundary, i.e. s.t. $x^2+y^2=1$. You can either argue via the Lagrangian, or for this particular function as $x^2=1-y^2$ you notice that the minimization problem reduces to $$\min_{y\in [-1,1]}1-y^2+(y-2)^2=\min_{y\in [-1,1]}1-y^2+(y-2)^2=\min_{y\in [-1,1]}5-4y$$ and it's quite clear that the minimum is achieved at $y=1$, wheres the maximum is at $y=-1$.