This equation showed up while I was studying an algorithm and even though it's simple-looking, it doesn't seem to be in the simple-to-solve category.
I'm not a mathematician so I did what every self-respecting engineer would do :
(1) I went to Wolfram Alpha : no success
(2) Used pen and paper to see if I would get an aha! moment : no success
(3) Went to my whiteboard thinking the outcome would be different : no comment
(4) Explored the equation graphically to figure out if and when it is solvable : success
(5) Used Maple to try to do what I wanted Wolfram Alpha to do : no success
Then I tried more serious things (please don't judge) :
(6) I tried to approximate the LHS of the equation with an exponential term hoping it would allow me to take one further step with the RHS : no success
(7) I sampled the constants randomly (1E3) and generated an array of numerical solutions to which I tried to fit different hyperplanes of some sorts : no success
(8) I sampled 1E6 solutions randomly and tried to see if I could make some statistical statement about them. The only thing that came out -- and this could simply be the result of the bounds I set on the variables -- is that the solution is often between 0 and 1, which isn't really surprising given the nature of the functions : no success
Things I thought about but didn't do because they provided no insight :
(8) Take the dataset generated in (7, 8) to train a neural net and see how good it could get while keeping a reasonably compact structure
---
There you have it. The reason why this is of interest to me is that I'm trying to make a statistical statement about the workings of a stochastic optimization algorithm and it would help to know more about the nature of the solution to this equation.
I would appreciate any sort of insights you might have, references you can point me to or magic tricks.
PS. I investigated the Lambert W function, because it " feels " like it could be what I'm looking for, but I'm still on that.
Thanks !
EDIT :
(1) As mentioned in the comments, the constants have to be positive with $t\,\epsilon\,[0,\infty)$ and $\alpha\,\epsilon\,(0,1)$
(2) Claude Leibovici has already did all of the leg work so far. The only thing still left to be figured out is the case when $0<\alpha<1$. Here is graphical proof that a solution exists in that case :
Example of graphical sln when $\alpha=0.9,\,A_{0},\,r=2,\,\gamma=0.5$
