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?

  • $\begingroup$ 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

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.

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

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