LinearProgramming returns an invalid null solution that apparently doesn't satisfy constraints without any warning

Question

Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later

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While searching to answer the question

• Solving the magic backyard puzzle with MMA

at some time I composed a set of arguments for LinearProgramming that apparently is not handled correctly. Running this code:

args = {cost, m, bs, vr, vd} = 
   Import["http://www.dropbox.com/s/73iz8t8ivakazvt/LinearProgrammingIssue.m?dl=1"];
sol = LinearProgramming @@ N@args

gives an invalid {0, 0, 0, ....., 0} solution that apparently doesn't satisfy the constraints

MapThread[#3[#1, #2] &, {m.sol, bs[[All, 1]], 
  bs[[All, 2]] /. {0 -> Equal, 1 -> GreaterEqual, -1 -> LessEqual}}]
Sign[m.sol - bs[[All, 1]]] == bs[[All, 2]]

{True, False, False, True, True, False, False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, False, False, False, False, False, False, False, False, False, False, False, False, False}

False

without raising any warning.

Is this a bug or am I missing something?

---

After @DanielLichtblau's comment, I also tried to reproduce with Minimize and Maximize.

args = {cost, m, bs, vr, vd} = 
   Import["http://www.dropbox.com/s/73iz8t8ivakazvt/LinearProgrammingIssue.m?dl=1"];

Module[{x, vars, constraints, objective, ranges, domains, N = N},
 vars = Array[x, Length@cost];
 objective = N@cost.vars;
 constraints = 
  MapThread[#3[#1, #2] &, {N@m.vars, N@bs[[All, 1]], 
    bs[[All, 2]] /. {0 -> Equal, 1 -> GreaterEqual, -1 -> LessEqual}}];
 ranges = MapThread[LessEqual @@ Riffle[#2, #1] &, {vars, vr}];
 domains = Thread[vars \[Element] N@vd];
 Minimize @@ {{objective, {constraints, ranges, domains}}, vars}
 ]

If I apply N as in LinearProgramming (so Module[{..., N=N}, ...]) the answer is immediate and still wrong.

If I don't apply N (so Module[{..., N=Identity}, ...]) Mathematica starts computing but I didn't have the patience to check if/when something happens.

I suspect this issue can be related to very big numbers in the cost vector (they are not machine precision Integer). They can converted into machine precision Real, but I'm not sure if a Real cost vector can be combined with Integer variables in a LinearProgramming/Minimize problem.