help in plotting Minimize function

Question

Please help in plotting the following function versus $c2 in [-1,1]$
$$S = mathop {min }limits_{theta 1,theta 2,phi 1,phi 2} F(c2,theta 1,theta 2,phi 1,phi 2)$$
where ${theta _i},{phi _i}$  belong to the interval $[0,2pi ]$.
Here my function:
S = 1/(32 Log[2]) E^(-I (ϕ1 + ϕ2)) (8 (-2 E^(I (ϕ1 + ϕ2)) + 
        1/2 √(E^(2 I (ϕ1 + ϕ2)) + 
            c2^2 (E^(2 I ϕ1) + E^(4 I ϕ1) + E^(
               2 I ϕ2) + E^(4 I ϕ2) + 
               3 E^(2 I (ϕ1 + ϕ2)) + E^(
               2 I (2 ϕ1 + ϕ2)) + E^(
               2 I (ϕ1 + 2 ϕ2))) + 
            c2 (1 + E^(2 I ϕ1) + E^(2 I ϕ2) + E^(
               4 I (ϕ1 + ϕ2)) + E^(
               2 I (2 ϕ1 + ϕ2)) + E^(
               2 I (ϕ1 + 2 ϕ2)))) Abs[Sin[2 θ1]] Abs[
          Sin[2 θ2]]) Log[
       1/8 E^(-I (ϕ1 + ϕ2)) (4 E^(
           I (ϕ1 + ϕ2)) - √(E^(
              2 I (ϕ1 + ϕ2)) + 
              c2^2 (E^(2 I ϕ1) + E^(4 I ϕ1) + E^(
                 2 I ϕ2) + E^(4 I ϕ2) + 
                 3 E^(2 I (ϕ1 + ϕ2)) + E^(
                 2 I (2 ϕ1 + ϕ2)) + E^(
                 2 I (ϕ1 + 2 ϕ2))) + 
              c2 (1 + E^(2 I ϕ1) + E^(2 I ϕ2) + E^(
                 4 I (ϕ1 + ϕ2)) + E^(
                 2 I (2 ϕ1 + ϕ2)) + E^(
                 2 I (ϕ1 + 2 ϕ2)))) Abs[
            Sin[2 θ1]] Abs[Sin[2 θ2]])] - 
     4 (4 E^(
         I (ϕ1 + ϕ2)) + √(E^(
            2 I (ϕ1 + ϕ2)) + 
            c2^2 (E^(2 I ϕ1) + E^(4 I ϕ1) + E^(
               2 I ϕ2) + E^(4 I ϕ2) + 
               3 E^(2 I (ϕ1 + ϕ2)) + E^(
               2 I (2 ϕ1 + ϕ2)) + E^(
               2 I (ϕ1 + 2 ϕ2))) + 
            c2 (1 + E^(2 I ϕ1) + E^(2 I ϕ2) + E^(
               4 I (ϕ1 + ϕ2)) + E^(
               2 I (2 ϕ1 + ϕ2)) + E^(
               2 I (ϕ1 + 2 ϕ2)))) Abs[Sin[2 θ1]] Abs[
          Sin[2 θ2]]) Log[
       1/8 E^(-I (ϕ1 + ϕ2)) (4 E^(
           I (ϕ1 + ϕ2)) + √(E^(
              2 I (ϕ1 + ϕ2)) + 
              c2^2 (E^(2 I ϕ1) + E^(4 I ϕ1) + E^(
                 2 I ϕ2) + E^(4 I ϕ2) + 
                 3 E^(2 I (ϕ1 + ϕ2)) + E^(
                 2 I (2 ϕ1 + ϕ2)) + E^(
                 2 I (ϕ1 + 2 ϕ2))) + 
              c2 (1 + E^(2 I ϕ1) + E^(2 I ϕ2) + E^(
                 4 I (ϕ1 + ϕ2)) + E^(
                 2 I (2 ϕ1 + ϕ2)) + E^(
                 2 I (ϕ1 + 2 ϕ2)))) Abs[
            Sin[2 θ1]] Abs[Sin[2 θ2]])]);

flinty · Accepted Answer

First make S into a proper function like this (clear kernel first):
S[ϕ1_?NumericQ, ϕ2_?NumericQ, θ1_?NumericQ, θ2_?NumericQ, c2_?NumericQ] := ...

Then write:
cx[a_] := 0 <= a <= 2 π
f[c2_] := NMinimize[{Re@S[ϕ1, ϕ2, θ1, θ2, c2], 
     cx[ϕ1], cx[ϕ2], cx[θ1], 
     cx[θ2]}, {ϕ1, ϕ2, θ1, θ2}][[1]] //Re

It only considers the real part as there's a very small imaginary part (floating point error?) which needs to be discarded.
Plot[f[c2], {c2, -1, 1}]

help in plotting Minimize function

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