How to deal with x/(W_{-1}(-x exp(-x)) +x) >= C, where x > 0 and W_{-1} denotes Lambert -W function

1

How to deal with x/( W_{-1}(-x exp(-x)) + x ) >= C, where x > 0 and W_{-1} denotes Lambert -W function, and C is a constant parameter.

2

Is C>0 or C<0 ? Can you prove the convexity of your constraint? You can find the lambert W-function at

3

I am not sure but

W(-x exp(-x)) = -x

holds or?

See Identities at https://en.wikipedia.org/wiki/Lambert_W_function

4

Thanks for your reply. Here, C > 0. I have proved that the constraint is convex, and the corresponding second derivative can be given by \frac{{W_{ - 1}}( - x e^{ - x} )}{x{( {1 + {W_{ - 1}}( { - x e^{ - x}} )} )^3}} > 0.

5

Thanks for your kindly reply. But I am afraid that W(-x exp(-x)) = -x is not holds. I verify it via Matlab.