Consider the minimization of $$f(x,y,z) = (x-\frac{y}{2})^2 + \frac{3}{4}(y-2)^2 +z^2 -3$$ on the set $$\{(x,y,z)\in\mathbb{R}^3|x^2+y^2\leq1\}$$
I'm trying to use the first and second order necessary conditions ($\nabla f \cdot v \geq 0$ and $v^T\nabla^2f v\geq 0$ for feasible $v$) to identify minimum points.
I think I've narrowed down the set of feasible $v$ to $v=\begin{bmatrix}u\\v\\w\end{bmatrix}$ such that $xu+yv <0$.
And I was able to show that $\nabla^2f(x,y,z)$ is positive definite (determinant of all leading minors are strictly positive).
However, I can't figure out how to generate a system of equations to actually identify any "critical" points.
Any suggestions would be appreciated. Thank you!
We need to look at first at the "interior points" s.t. $x^2+y^2<1$ by the condition $\nabla f=0$.
I've found no critical interior points.
Then we need to check the points $x^2+y^2=1$ using directly that condition.
That should lead to the minimization of $g(x,y)=xy+\frac32 y$ under the given condition.