I am currently trying to apply the FindMinimum function to find the lowest energy of an expression. My current expression as of now is relatively complicated. It is the summation of 48 similar terms, and after simplifying one of these terms, I get the expression
((Cot[ϕ[1]] Cot[ϕ[6]/2] - Cot[ϕ[6]] Csc[ϕ[1]] - Csc[ϕ[1]] Csc[ϕ[6]] - I
Cos[θ[6]] Sin[θ[1]] + Cos[θ[1]] (Cos[θ[6]] + I Sin[θ[6]]) +
Sin[θ[1]] Sin[θ[6]]) (Cot[ϕ[5]] Cot[ϕ[6]/2] - Cot[ϕ[6]] Csc[ϕ[5]] -
Csc[ϕ[5]] Csc[ϕ[6]] + I Cos[θ[6]] Sin[θ[5]] + Cos[θ[5]] (Cos[θ[6]] - I Sin[θ[6]]) +
Sin[θ[5]] Sin[θ[6]]) (-Cot[ϕ[2]] Csc[ϕ[1]] + Csc[ϕ[1]] Csc[ϕ[2]] +
I Cos[θ[2]] Sin[θ[1]] + Cos[θ[1]] (Cos[θ[2]] - I Sin[θ[2]]) +
Sin[θ[1]] Sin[θ[2]] - Cot[ϕ[1]] Tan[ϕ[2]/2]) (-Cot[ϕ[3]] Csc[ϕ[2]] + Csc[ϕ[2]] Csc[ϕ[3]] + I Cos[θ[3]] Sin[θ[2]] + Cos[θ[2]] (Cos[θ[3]] - I Sin[θ[3]]) +
Sin[θ[2]] Sin[θ[3]] - Cot[ϕ[2]] Tan[ϕ[3]/2]) (-Cot[ϕ[4]] Csc[ϕ[
3]] + Csc[ϕ[3]] Csc[ϕ[4]] + I Cos[θ[4]] Sin[θ[3]] +
Cos[θ[3]] (Cos[θ[4]] - I Sin[θ[4]]) + Sin[θ[3]] Sin[θ[4]] -
Cot[ϕ[3]] Tan[ϕ[4]/2]) (-Cot[ϕ[5]] Csc[ϕ[4]] + Csc[ϕ[4]] Csc[ϕ[5]] +
I Cos[θ[5]] Sin[θ[4]] + Cos[θ[4]] (Cos[θ[5]] - I Sin[θ[5]]) +
Sin[θ[4]] Sin[θ[5]] - Cot[ϕ[4]] Tan[ϕ[5]/2]))/(√((1 +
Abs[Cot[ϕ[6]/2] (Cos[θ[6]] - I Sin[θ[6]])]^2) (1 +
Abs[Cot[ϕ[6]/2] (Cos[θ[6]] + I Sin[θ[6]])]^2) (1 +
Abs[(-Cos[θ[1]] + I Sin[θ[1]]) Tan[ϕ[1]/2]]^2) (1 +
Abs[(Cos[θ[1]] + I Sin[θ[1]]) Tan[ϕ[1]/2]]^2) (1 +
Abs[(-Cos[θ[2]] + I Sin[θ[2]]) Tan[ϕ[2]/2]]^2) (1 +
Abs[(Cos[θ[2]] + I Sin[θ[2]]) Tan[ϕ[2]/2]]^2) (1 +
Abs[(-Cos[θ[3]] + I Sin[θ[3]]) Tan[ϕ[3]/2]]^2) (1 +
Abs[(Cos[θ[3]] + I Sin[θ[3]]) Tan[ϕ[3]/2]]^2) (1 +
Abs[(-Cos[θ[4]] + I Sin[θ[4]]) Tan[ϕ[4]/2]]^2) (1 +
Abs[(Cos[θ[4]] + I Sin[θ[4]]) Tan[ϕ[4]/2]]^2) (1 +
Abs[(-Cos[θ[5]] + I Sin[θ[5]]) Tan[ϕ[5]/2]]^2) (1 +
Abs[(Cos[θ[5]] + I Sin[θ[5]]) Tan[ϕ[5]/2]]^2)))
where I have included in the input code for it in Mathematica. The $\phi[i]$ and $\theta[i]$ are simply variables and the index $i$ ranges in $\{1..6\}$. I obtained an analogous version of this expression by simplifying with the assumptions that $0 \leq \phi[i] \leq 2*\pi,\quad 0 \leq \theta[i] \leq \pi$, however this took longer, and the expression didn't seem that much simpler.
My question is that since I'm trying to apply the FindMinimum function to a summation over 48 expressions, each of which are analogous to the expression above, should I bother simplifying the sum before applying the FindMinimum function to it? Are there any other things I could do to ease the expression for FindMinimum?
In addition, I know the minimum exists for the summation, and I'm trying to simultaneously find the minimum of this plus a much simpler expression. Hence, I was planning to use the built-in "goal programming" https://reference.wolfram.com/language/tutorial/ConstrainedOptimizationLocalNumerical.html with the added constraints of the intervals on the variables as mentioned above.
If it helps, I could provide the expression for the first term, before simplification to the above, purely in terms of $\theta$ and $\phi$. The above mess of trigonometric functions comes from transforming those coordinates to euclidean coordinates. As a note, one must fix values for $\theta$ and $\phi$ for one index $i$, before performing FindMinimum, so that one does not get infinite configurations of the same value.
Thanks!
FullSimplify[Abs[Cot[ϕ[6]/2] (Cos[θ[6]] - I Sin[θ[6]])], θ[6] ∈ Reals]
yieldsAbs[Cot[ϕ[6]/2]]
, which may be useful. $\endgroup$