Conic Representation of half-sigmoid

[Sam Chevalier]

The sigmoid function is nonconvex: 1/(1+e^-x).

I am searching for a conic expression for the "right half" of the sigmoid: t \le 1/(1+e^-x) , x>0. This is the shaded blue region with a red dot. I suspect there is a conic expression, because I can fit a modified (see equation below, and green line) Softplus (which does have a conic reformulation) "almost" perfectly by tuning some parameters, but the slope is never quite right.

Any ideas MOSEK?

Thanks!

Sam

[Michal Adamaszek]

Unlikely  to be conic representable in any natural way according to the expert here:

https://ask.cvxr.com/t/is-1-1-exp-x-solvable-here-x-0/724

This is a common problem with functions that switch curvature through an inflection point, as opposed to changing curvature through a vertical asymptote / being undefined,  like 1/x in x=0, in which case there is more chance to make it work.

Michal

[Sam Chevalier]

Thanks, Michal! That makes sense, and I guess is what I expected.

Sam