2
$\begingroup$

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$

where $x_{i}, y_{i} \in \left(0, 1\right)$ are constants, $w\in \mathbb{R}$. Can you prove or disprove it?

$\endgroup$
1
  • $\begingroup$ Interestingly, this is like fitting a logistic function via least squares. My guess is that there is something on this in mathematical statistics. $\endgroup$
    – Nameless
    Dec 10 '13 at 19:20
5
+100
$\begingroup$

The way it is phrased, the conjecture is false. Consider the example $$f(w)=1/2\left[\left(\frac{1}{1+e^{-0.1w}}-0.9\right)^2+\left(\frac{1}{1+e^{-0.9w}}-0.1\right)^2\right],$$ or plotted: enter image description here

Clearly, there is a unique maximum and a unique minimum, i.e., there are two extrema.

$\endgroup$
1
  • $\begingroup$ Thanks for your counterexample. $\endgroup$ Dec 10 '13 at 19:29

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.