How to couple NIntegrate with FindMinValue

Question

I have a problem with a correct transfer of values of parameters when NIntegrate (and some subsequent functions) are coupled with minimizing procedures in Mathematica. Namely, let's define an operator:

V[r_,R_] := If[r < R, 0., 1.]
H[r_, R_] := m* D[#, r] + V[r, R]*# &;

which acts on a function (the function below is only an example but their nested form and 'ifs', which are important to me, would be possibly crucial)

X[R_, b_?NumberQ] := -b / Sin[b*R];
F1[r_, R_, b_?NumberQ] := If[r<=R, Sin[b*r] / r, Exp[-X[R, b]*r] / r]
F[r_, R_, b_?NumericQ] := F1[r, R, b]
m = 1.0;

I try to minimize this numerical integral (the best method seems to be 'FindMinValue' because initial value of minimizing parameter can be specified by hand). The two commands below do calculate it but non-numerical values are indicated in messages; how to avoid them?

(*1*) FindMinValue[NIntegrate[F[r,2.5,b]*H[r, 2.5]F[r,2.5,b]],{r,1,2.5}], {b,1.0}] (* R=2.5 is set here by hand *)
(*2*) With[{R=2.5},FindMinValue[NIntegrate[F[r,R,b]*H[r, R][F[r,R,b]],{r,1,R}],{b, 1.0}]]

But the following does not return any messages, surprisingly. What is the difference in transfer of 'R' into 'FindMinValue' here and in those previous cases? I cannot see a general rule.

(*3*) Do[{R = 2.5*i,abc = FindMinValue[NIntegrate[F[r,R,b]*H[r, R][F[r,R,b]],{r,1,R}],{b,1.0}],Print[abc]},{i,2}]

I would be grateful for help.

Answer 1

1. Your F wrapper function is doing nothing for you in your code, so I removed it and replaced it with direct calls to F1.
2. NIntegrate in the argument to FindMinValue should not be evaluated unless it is passed explicitly numerical arguments, so it is best to wrap it in a function protected by NumericQ (functiontominimize below). Since all the other functions you defined are called from this one, they don't really need to include NumericQ checks.

3. The minimization was running into numerical trouble. FindMinValue suggested to use more numerical precision, so I substituted your approximate values with exact ones, and increased the WorkingPrecision.

   Clear[V, H, X, F1]
   
   V[r_, R_] := If[r < R, 0., 1.]
   H[r_, R_] := m*D[#, r] + V[r, R]*# &;
   X[R_, b_] := -b/Sin[b*R];
   F1[r_, R_, b_] := If[r <= R, Sin[b*r]/r, Exp[-X[R, b]*r]/r]
   m = 1;
   
   functiontominimize[r_, R_, b_?NumericQ] := 
        NIntegrate[F1[r, R, b]*H[r, R][F1[r, R, b]], {r, 1, R}]
   
   With[{R = 25/10}, 
        FindMinValue[functiontominimize[r, R, b], b, WorkingPrecision -> 20]
   ]
   
   (* Out: -0.47610029093305050729 *)
   

The code above returns no message and executes faster as well.