Why minimization does not work with symbolic array as arguments

Question

If I try to minimize with constraints a function of several variables, with Gamma regularized function involved in the constraints it seems to works, as shown below (this is just a dummy example problem to try), where S, A , β and m are constants previously declared and assigned a value:

    Minimize[-Log[v] - Log[y] - Log[z], 
 Gamma[m, S*A*400^β*m*v^(1/m)]/Gamma[m] + 
    Gamma[m, S*A*400^β*m*z^(1/m)]/Gamma[m] + 
    Gamma[m, S*A*400^β*m*y^(1/m)]/Gamma[m] <= 1 && v <= 0.1 && 
  v >= 0.001 && z <= 0.1 && z >= 0.001 && y <= 0.1 && y >= 0.001, {v, 
  y, z}]
    {6.90776, {v -> 0.1, y -> 0.1, z -> 0.1}}

Now, since I need later to solve a problem with hundreds of variables, I try the same problem with an array of variables, but it does not work:

 Minimize[{\!\(
\*UnderoverscriptBox[\(∑\), \(i = 1\), \(3\)]\(-Log[h[i]]\)\), 
  Join[{\!\(
\*UnderoverscriptBox[\(∑\), \(i = 1\), \(3\)]\(GammaRegularized[
        1, S*A*
\*SuperscriptBox[\(400\), \(β\)]*m*h[i]]\)\) <= 
      1}, {Table[h[i] <= 0.1, {i, 3}] /. 
      List -> And} , {Table[h[i] >= 0.001, {i, 3}] /. 
      List -> And}] /. List -> And}, Table[h[i], {i, 3}]]

Minimize[{-Log[h[1]] - Log[h[2]] - Log[h[3]], 
  E^(-123.319 h[1.]) + E^(-123.319 h[2.]) + E^(-123.319 h[3.]) <= 1 &&
    h[1] <= 0.1 && h[2] <= 0.1 && h[3] <= 0.1 && h[1] >= 0.001 && 
   h[2] >= 0.001 && h[3] >= 0.001}, {h[1], h[2], h[3]}]

After trying several options, it seems that if I rewrite the problem using only variables h[i], instead of the expression in the argument to the GammarRegularizedfunction it works again (see below):

Minimize[{\!\(
\*UnderoverscriptBox[\(∑\), \(i = 1\), \(3\)]\(-Log[
\*FractionBox[\(h[i]\), \(S*A*
\*SuperscriptBox[\(400\), \(β\)]*m\)]]\)\), Join[{\!\(
\*UnderoverscriptBox[\(∑\), \(i = 1\), \(3\)]\(GammaRegularized[
        1, h[i]]\)\) <= 
      1}, {Table[h[i] <= S*A*400^β*m*0.1, {i, 3}] /. 
      List -> And} , {Table[
       h[i] >= S*A*400^β*m*0.001, {i, 3}] /. List -> And}] /. 
   List -> And}, Table[h[i], {i, 3}]]

{6.90776, {h[1] -> 12.3319, h[2] -> 12.3319, h[3] -> 12.3319}}

Why is it that I cannot use a expression as argument? I would need to pass expressions in my real problem as argument.

Answer 1

Mathematica does not like your h[i]s.

If you introduce

 shrt[head_, 
       indices_] := 
     ToExpression[StringJoin[
         ToString /@ Flatten[
             {head, 
               indices}]]]

so that e.g. shrt[h,1] becomes h1 or shrt[h,{i,j}] hij

Then this works:

S = 10^-9.2;
cl = 3*10^8 ;
f = 5.9*10^9;
w = cl/f ;
A = (4*π/w)^(2/1);
β = 2.5 ;
m = 1 ;

Join[{Sum[
   GammaRegularized[1, S*A*400^β*m*shrt[h, i]] , {i, 
    3}] <= 
        1}, {Table[shrt[h, i] <= 0.1, {i, 3}] /. 
        List -> And} , {Table[shrt[h, i] >= 0.001, {i, 3}] /. 
        List -> And}] /. List -> And}, Table[shrt[h, i], {i, 3}]]

(* {6.90776,{h1->0.1,h2->0.1,h3->0.1}} *)