I'd like to minimize this $\mathbb{R}^4$ function: \begin{equation} f(x,y,w,z) = xw+yz \end{equation} subject to the constraint \begin{equation} x^2+y^2-w^2-z^2=0 \end{equation} Well, when I apply the Lagrange multiplier method to the problem, I get as the only real solution the point $(0,0,0,0)$ which is coherent with the constraint but actually the point $(k,0,-k,0)$ with real $k$ is a much better minimum! I feel that since the method gives the stationary points of the lagrangian it can fail to localize 'minimizing' manifolds such as $x=-w$ in this particular case.
Am I right? What is the best way to understand what goes on in this particular problem?
This is a function that is unbounded.
$$f(k,0,-k,0)=-k^2$$
and we can take $k$ to be arbitrary large number and hence there is no global minimum.
The function can also be made arbitrarily large.
$$f(k,0,k,0)=k^2$$
There is no global maximum as well.
This is a non-convex function.
To understand the problem better, we can study a similar problem of $g(x,y)=xy$ subject to $x^2=y^2$ and we can see a saddle point and the function grows arbitarily large in one direction and become arbitrarily small in another.
Notice that a stationary point doesn't mean a point must be a local minimum or global maximum, an example would be $f(x)=x^3$, the only stationary point is $0$, but it is neither a maximum not minimum.