Findminimum of Integral

Question

I am trying to find the value of $u$ that minimizes the following expression:

q = 1 - Exp[-u/b];
expr = Log[(1/(2 + b)) (1/(1 - q + q^2)) (q + 
       q^2 + (1 - q)^2 Exp[-u/2])/0.3553];   (*SEC BER equation *)
FindMinimum[Integrate[expr, {b, 1, 10}], {u, 0, 10}]

However, it does not yield any output. What's wrong with the code?

Answer 1

You can do something like this.

Clear[expr];
q = 1 - Exp[-u/b];
expr[u_?NumericQ] := 
 NIntegrate[
  Log[(1/(2 + b)) (1/(1 - q + q^2)) (q + q^2 + (1 - q)^2 Exp[-u/2])/0.3553], {b, 1, 10}];

Now easy to optimize.

FindMinimum[expr[u], {u, 0, 10}]

{-10.5871, {u -> 1.2105}}