1
I am not sure, but I think we can do
$$
\text{minimize}~ \|\frac 1 {1+e^{-X\beta}} -y \|_2^2+ \lambda\|\beta\|_1
$$
Where $X$ is the data matrix and $y$ is the response and $\beta$ is the coefficients. The objective is convex.
And
$$ 0< \frac 1 {1+e^{-X\beta}} < 1$$