$x$ is a vector that $x=(x_1,\ldots,x_n)$

$f(x) = -\sum_{i=1}^n \log(x_i)$

I know this is equivalent of proving that $\Pi x_{i}$ is a convex set or not, but how to prove this step? Since it seems that I can't write derivative from this, or there are other ways that I can escape from proving the convexity of the function I wrote above?

  • $\begingroup$ Is $x$ a vector? Is $x=(x_1,\ldots,x_n)$? $\endgroup$ – ajotatxe Sep 7 '16 at 23:01
  • $\begingroup$ It is, thank you and I've updated my question. $\endgroup$ – good2know Sep 7 '16 at 23:03

$f$ is defined on $\{x: x_i > 0\}$ and is of class $C^{\infty}$ there. Now the Hessian matrix: $$H_f(x) = \text{diag}\left( \frac1{x_1^2}, \ldots, \frac1{x_n^2}\right)$$

is clearly positive semi-definite everywhere. So the function is convex.


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.