I have a complicated (looking) equation Gq, that I'm looking to minimize:
Gq = {b \[Gamma] - (E^((b Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2])/\[Sigma]^2) (E^((b Sqrt[\[Mu]^2 \
+ 2 \[Lambda] \[Sigma]^2])/\[Sigma]^2) mc +
E^((b \[Mu])/\[Sigma]^2) me) ((E^((b Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2])/(2 \[Sigma]^2)) Sqrt[
E^((b Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2])/\[Sigma]^2) mc +
E^((b \[Mu])/\[Sigma]^2) me])/
Sqrt[mc +
E^((b (\[Mu] +
Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2]))/\[Sigma]^2) me])^(-1 - \
\[Mu]/Sqrt[\[Mu]^2 + 2 \[Lambda] \[Sigma]^2]) (-\[Mu]^2 -
2 \[Lambda] \[Sigma]^2 + \[Mu] Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2]))/((-1 +
E^((2 b Sqrt[\[Mu]^2 +
2 \[Lambda] \[Sigma]^2])/\[Sigma]^2)) \[Lambda] \
(-\[Mu] + Sqrt[\[Mu]^2 + 2 \[Lambda] \[Sigma]^2]))}
I tried Minimize[Gq,b] directly but it just hangs. Then I tried solve D by taking the derivative and setting to zero (it's a convex function), including all of the positivity and non-negativity conditions that should make it easier to solve.
Solve[D[Gq, b] == 0 && b > 0 && \[Lambda] > 0 && \[Sigma] > 0 && me > 0 && mc > 0, b]
But it still continues to hang and doesn't compute. Your help would be VERY much appreciated. Even when I continue to substitute more direct values
me == 0.005 && mc = 0.0075 && \[Lambda] == 0.1 && \[Sigma] > 10000 && \[Mu] == -10000
so that the only free variable is b, it still doesn't solve it.