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I have a family of equations I want to solve over the non-negative integers. One particular instance of it is this.

A = {{1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}}

b = {1, 2, 3, 1, 2, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0}

I'd like to solve the problem $$ A.x = b,\quad x \in \mathbb{N}^{22} $$ So I call up LinearProgramming and ask

LinearProgramming[ Table[0, {22}], A, {#,0}& /@ b, 
                   Table[{0,Infinity}, {22}],Table[ Integers, {22}]]

The purported solution is

c = {1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

which is not, actually, a solution of the problem.

Why does this happen? Why doesn't Mathematica tell me that there is no solution?

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  • 1
    $\begingroup$ Looks like a bug. Will investigate.. $\endgroup$ – Daniel Lichtblau Oct 28 '13 at 15:08
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This bug has been fixed in the recently released version 12.0

LinearProgramming[Table[0,{22}], A, {#, 0}& /@ b,                                                       
       Table[{0, Infinity}, {22}],Table[Integers,{22}]] // Head

LinearProgramming::lpip: Warning: integer linear programming will use a machine-precision approximation of the inputs.

LinearProgramming::lpsnf: No solution can be found that satisfies the constraints.

(* LinearProgramming *)
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  • $\begingroup$ That fixed earlier, at least in version 11.3. $\endgroup$ – user64494 Apr 17 '19 at 6:21
  • $\begingroup$ Not exactly, it also depended on the operating system. $\endgroup$ – ilian Apr 17 '19 at 12:28

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