Minimization of a summation of two functions with respect to two constrained variables

Question

I have a question regarding the minimization of a function, which is a summation of two separate functions, with a discrete list for one of the variables.

$S[a,c]= \sqrt{\sum_{b=0,4,8,12,16}{(f[b]-g[b,c,a])^2}} $

---

$f[b]$ is linear function with one variable , However $g[b,c,a]$ is a complex number function

and note that there are also constraints, $0<c<1$ and $0<a<5$

My question is, how do I first perform the summation then minimize the function $S[a,c]$ to optimize the variable $a$ and $c$ on the function.

Is there anyway I can used the NMinimize option?

I tried using this built in operator and it gives me an error which says that the objects are of unequal length.

Any help regarding this problem would be appreciated.

Answer 1

This not an answer but extended comment. Try something like this.

f[b_] := 2 b - 5
g[b_, c_, a_] := b^2 + 2 c^4 + a^4 + 2
S[a_?NumberQ, c_?NumberQ] := Sqrt[
 Total@Table[(f[b] - g[b, c, a])^2, {b, 0, 16, 4}]]

NMinimize[{S[a, c], 0 < c < 1, 0 < a < 5}, {a, c}]

{269.794, {a -> 0.00163593, c -> 0.00210146}}