1
$\begingroup$

I try to find the vector $x = (x_1, \cdots, x_n)$ to maximize function $f(x)=f(x_1, \cdots, x_n)$ subject to the constraint $x_1^2 + \cdots +x_n^2 = a$, where $a$ is a positive constant. I use Lagrange multilplier and show that $df/dx_1=0$ at $x_1 = 0$. So can I conclude that the optimal solution $x$ must has the form $x = (0, x_2, \cdots, x_n)$?

$\endgroup$
  • $\begingroup$ The solutions of an optimization problem are usually NOT critical points of $f$. $\endgroup$ – Taladris May 28 '14 at 3:39
2
$\begingroup$

The FOC conditions you get are $df/dx_i-2\lambda x_i=0$. This does not mean that $\dfrac{df}{dx_i}=0$ at $x_i=0$.

Of course, if you are able to prove that $\frac{df}{dx_1}(0,x_2,\ldots,x_n)=0$ then you can test if there is a solution $x=(0,x_2,\ldots,x_n)$.

If $x_i\neq 0$ for all $i$ you can use $\dfrac{df/dx_j}{df/dx_i}=x_j/x_i$ and $x_1^2+\ldots+x_n^2=a$ and solve for the $x$s.

$\endgroup$

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.