I want to find the maximum of

d2= (Sqrt[Abs[a1^2 b1^2]] + Sqrt[Abs[a[2]^2 b[2]^2]] + Sqrt[Abs[a[4]^2 b[4]^2]] + Sqrt[Abs[a[5]^2 b[5]^2]] + Sqrt[Abs[a[6]^2 b[6]^2]] + Sqrt[Abs[a[7]^2 b[7]^2]] + (1/Sqrt[2])([Sqrt]Abs[a[3]^2 b[3]^2 + a[8]^2 b[8]^2 + a[9]^2 b[9]^2 + a[10]^2 b[10]^2 – [Sqrt](((a[3] b[3] + a[8] b[8])^2 + (a[9] b[9] – a[10] b[10])^2) ((a[3] b[3] – a[8] b[8])^2 + (a[9] b[9] +a[10] b[10])^2))]) + (1/Sqrt[2])([Sqrt]Abs[a[3]^2 b[3]^2 + a[8]^2 b[8]^2 + a[9]^2 b[9]^2 + a[10]^2 b[10]^2 + [Sqrt](((a[3] b[3] + a[8] b[8])^2 + (a[9] b[9] – a[10] b[10])^2) ((a[3] b[3] – a[8] b[8])^2 + (a[9] b[9] +a[10] b[10])^2))]))^2

subject to a very large set of constraints designated “vcf” in the notebook

https://www.wolframcloud.com/obj/slater/Published/MaximizeNorms

The command

```
G1 = NMaximize[{d2, vcf}, Join[Array[a, 10], Array[b, 10]]]
```

yields the result

```
{0.1565, {a[1] -> -0.217171, a[2] -> 0.228215, a[3] -> 0.242746, a[4] -> 0.185082, a[5] -> -0.194495, a[6] -> -0.239223, a[7] -> -0.25139, a[8] -> -0.236464, a[9] -> 0.157249, a[10] -> 0.152502, b[1] -> -0.187483, b[2] -> 0.197017, b[3] -> 0.217313, b[4] -> 0.201103, b[5] -> 0.19137,b[6] -> 0.269263, b[7] -> -0.256231, b[8] -> 0.197663, b[9] -> 0.0729522, b[10] -> 0.17673}}
```

which does in fact satisfy the constraint vcf.

However, the command

```
G2 = NMaximize[{d2, vcf}, Join[Array[a, 10], Array[b, 10]], Method -> "DifferentialEvolution"]
```

yields the larger (and more plausible/attractive) result

```
{0.186769, {a[1] -> -0.268552, a[2] -> -0.255555, a[3] -> 0.227544,a[4] -> -0.18653, a[5] -> 0.196016, a[6] -> -0.203221, a[7] -> 0.193385, a[8] -> -0.316546, a[9] -> -0.162042, a[10] -> -0.153076, b[1] -> -0.237133, b[2] -> -0.249411, b[3] -> 0.215763, b[4] -> 0.20219, b[5] -> -0.192236, b[6] -> -0.193805, b[7] -> 0.203839, b[8] -> -0.31299, b[9] -> -0.026066, b[10] -> -0.201624}}
```

but does not fully satisfy the constraint vcf.

(I believe that there is an exact value–possibly rational in nature [not too much larger than the 0.186769]–for this [quantum-information-theoretic] problem, but it certainly seems to be too demanding a task to obtain it.)