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?

  • $\begingroup$ I guess the problem is that the constraint set is not bounded... $\endgroup$ – marco trevi Oct 20 '17 at 19:34

This is a function that is unbounded.


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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.