Constraints and Tolerance

So I’m running NMaximize for optimizing a value and for constraints, I need the parameters to belong to a discrete set of elements. Basically, the constraint looks like,

And @@Table[{Subscript[x, i], Subscript[y, i]} \[Element] Table[Subscript[e, i, n], {i, 1, n}],{i, 1, k}] 

,where {Subscript[x, i], Subscript[y, i]} are my parameters satisfying the constraint that they should belong to Table[Subscript[ex, i, n], {i, 1, n}]

But NMaximize do not consider this as a constraint. Then I changed the constraint to the convex hull of Table[Subscript[ex, i, n], {i, 1, n}] with an additional constraint to pick out the extreme points aka the vertices. Now my code looks like,

And @@ Table[{Subscript[x, i], Subscript[y, i]} \[Element]  ConvexHullMesh[Table[Subscript[ex, i, n], {i, 1, n}]] &&  Subscript[z, i] == 1/2 &&  Subscript[x, i]^2 + Subscript[y, i]^2 == Subscript[r, n]/2, {i, 1,  k}] 

But, when I run this it outputs this error

Obtained solution does not satisfy the following constraints within
Tolerance -> 0.001`

What do I do?

FULL CODE

Subscript[r, n_] := Sqrt[Sec[Pi/n]]; Subscript[w, i_,  n_] := {Subscript[r, n] Cos[2 Pi i/n],  Subscript[r, n] Sin[2 Pi i/n], 1}; Subscript[e, i_, n_] :=  1/2 {Subscript[r, n] Cos[(2 i - 1) Pi/n],  Subscript[r, n] Sin[(2 i - 1) Pi/n], 1}; Subscript[ex, i_, n_] :=  1/2 {Subscript[r, n] Cos[(2 i - 1) Pi/n],  Subscript[r, n] Sin[(2 i - 1) Pi/n]}; u = {0, 0, 1}; f = (u - #) &; (*Factors={1,2,3,12,13,23,123}*)  Factors = Times @@@ Subsets[Transpose@Tuples[{1, -1}, 3], {1, 3}]; (*Rearrange*) (*Rearrange the numbers in the RHS to obtain different \ combinations*)  Factors[[{1, 2, 3, 4, 5, 6, 7}]] = Factors[[{1, 2, 3, 4, 5, 6, 7}]]; Factors = Transpose[Factors]; Vec[j_] := {Subscript[x, j], Subscript[y, j], Subscript[z, j]}; AllParameters[k_] :=  Module[{i},  Flatten[Table[{Subscript[x, i], Subscript[y, i], Subscript[z,    i]}, {i, 1, k}]]]; AllConstraints[n_, k_] :=  Module[{i},  And @@ Table[{Subscript[x, i], Subscript[y, i]} \[Element]     ConvexHullMesh[Table[Subscript[ex, i, n], {i, 1, n}]] &&    Subscript[z, i] == 1/2 &&    Subscript[x, i]^2 + Subscript[y, i]^2 == Subscript[r, n]/2, {i,    1, k}]];       GPT[n_, k_] := Module[{ro, co, ve, i},  FunFactor = Factors[[1 ;; 8, 1 ;; k]] /. {1 -> Identity, -1 -> f};  vec = Table[Subscript[v,   i], {i, 1, k}] /. {Subscript[v, j_] -> Vec[j]};  vecs = Table[  Total[Table[FunFactor[[ro, co]][vec[[co]]], {co, 1, k}]], {ro, 1,   8}];  max = Total[  Table[Max[   Map[vecs[[ve]].# &, Table[Subscript[w, i, n], {i, 1, n}]]], {ve,    1, 8}]];  {time, out} =   Timing[NMaximize[{max, AllConstraints[n, k]}, AllParameters[k],   Method -> "NelderMead"]];  Print[out, out[[1]]/(k 8), "   ", time]; {time, out} =   Timing[NMaximize[{max, AllConstraints[n, k]}, AllParameters[k],   Method -> "DifferentialEvolution"]];  Print[out, out[[1]]/(k 8), "   ", time]; {time, out} =   Timing[NMaximize[{max, AllConstraints[n, k]}, AllParameters[k],   Method -> "SimulatedAnnealing"]];  Print[out, out[[1]]/(k 8), "   ", time];]       GPT[4, 7]