NMinimize::bcons when making the conditions only slightly more complex

Question

I am trying to do (global) integer constrained optimization using NMinimize. I have some polynomial consisting of many variables $x_{1,1}...x_{5,8}$ and a couple of constraints $A, B, ..., F$. I am having problems with a particular constraint. Unfortunately I am not able to create a minimal working example, as it seems that the problem is related to the complexity of the calculation.

Let's say that

A = 
-800.  + x[1, 1] + x[2, 1] + x[3, 1] + x[4, 1] + x[5, 1] >= 0 && 
-500.  + x[1, 2] + x[2, 2] + x[3, 2] + x[4, 2] + x[5, 2] >= 0 && 
-2000. + x[1, 3] + x[2, 3] + x[3, 3] + x[4, 3] + x[5, 3] >= 0 && 
-5000. + x[1, 4] + x[2, 4] + x[3, 4] + x[4, 4] + x[5, 4] >= 0 && 
-1000. + x[1, 5] + x[2, 5] + x[3, 5] + x[4, 5] + x[5, 5] >= 0 && 
-6000. + x[1, 6] + x[2, 6] + x[3, 6] + x[4, 6] + x[5, 6] >= 0 && 
-1000. + x[1, 7] + x[2, 7] + x[3, 7] + x[4, 7] + x[5, 7] >= 0 && 
-4000. + x[1, 8] + x[2, 8] + x[3, 8] + x[4, 8] + x[5, 8] >= 0

Then, Mathematica successfully solves the problem

NMinimize[{Func, A && B && C && D && E && F}, Flatten@X];

Now, if I make the one constraint A more complex by allowing also 0 for each row, I am getting

NMinimize::bcons: The following constraints are not valid: ... Constraints should be equalities, inequalities, or domain specifications involving the variables.

A then reads:

A = 
(x[1, 1] + x[2, 1] + x[3, 1] + x[4, 1] + x[5, 1] >= 800. || 
 x[1, 1] + x[2, 1] + x[3, 1] + x[4, 1] + x[5, 1] == 0) && 
(x[1, 2] + x[2, 2] + x[3, 2] + x[4, 2] + x[5, 2] >= 500. || 
 x[1, 2] + x[2, 2] + x[3, 2] + x[4, 2] + x[5, 2] == 0) && 
(x[1, 3] + x[2, 3] + x[3, 3] + x[4, 3] + x[5, 3] >= 2000. ||
 x[1, 3] + x[2, 3] + x[3, 3] + x[4, 3] + x[5, 3] == 0) && 
(x[1, 4] + x[2, 4] + x[3, 4] + x[4, 4] + x[5, 4] >= 5000. ||
 x[1, 4] + x[2, 4] + x[3, 4] + x[4, 4] + x[5, 4] == 0) && 
(x[1, 5] + x[2, 5] + x[3, 5] + x[4, 5] + x[5, 5] >= 1000. ||
 x[1, 5] + x[2, 5] + x[3, 5] + x[4, 5] + x[5, 5] == 0) && 
(x[1, 6] + x[2, 6] + x[3, 6] + x[4, 6] + x[5, 6] >= 6000. ||
 x[1, 6] + x[2, 6] + x[3, 6] + x[4, 6] + x[5, 6] == 0) && 
(x[1, 7] + x[2, 7] + x[3, 7] + x[4, 7] + x[5, 7] >= 1000. ||
 x[1, 7] + x[2, 7] + x[3, 7] + x[4, 7] + x[5, 7] == 0) && 
(x[1, 8] + x[2, 8] + x[3, 8] + x[4, 8] + x[5, 8] >= 4000. ||
 x[1, 8] + x[2, 8] + x[3, 8] + x[4, 8] + x[5, 8] == 0)

I checked that it works with the condition A alone, i.e., without B...F, so it's not syntax or anything. Also, this second version of A would also allow solutions of the first version. What causes my error here?

---

Edit: I tried to remove stuff from the notebook step-by-step while keeping the error. I believe the code below is a minimal working example. I hope, the error's origin really is the same as in my original problem.

X = Array[Subscript[x, ##] &, {3, 3}];

s = {{0.3`,0.3`,1.2`},{1.2`,1.2`,0.3`},{1.4`,1.4`,1.4`}};
b = {10000.`,5000.`,10000.`};
m = {800.`,500.`,2000.`};

MinOr[xx_,mm_] = xx >= mm || xx == 0;

C1 = AllTrue[MapThread[MinOr,{Total[X,1],m}], # &];
C2 = AllTrue[Total[Transpose[X],1]-b, # == 0 &];
C3 = AllTrue[Flatten@X, # >= 0 &];
C4 = AllTrue[Flatten@X,Element[#, Integers] &];

C12 = AllTrue[Total[X, 1] - m, # >= 0 &];

NMinimize[{Total[Flatten[X*s]], C1 && C2 && C3 && C4},Flatten@X]

Replacing C1 with C12 Mathematica is able to solve it, but not with C1.

Answer 1

Mathematica seems to choke on the fact the some of the parts of the logical expansion C1&&C2&&C3&&C4 are pathological. In particular, when you define

cons = LogicalExpand[C1 && C2 && C3 && C4]

then

NMinimize[{Total[Flatten[X*s]], cons[[1]]},Flatten@X]

produces an error message. cons[[1]] is case that all MinOr options are set to zero. This case is clearly pathological since the combination with C3 means that all x must vanish. (Which may be hard for Mathematica to figure out without at total Reduce of the constraints.)

Mathematica's problem in this case can be circumvented by introducing a new constraint

C0 = AllTrue[Flatten[X], (# == 0 &)]

and using it explicitly resolve the pathological constraint:

NMinimize[
 {Total[Flatten[X*s]], C0||LogicalExpand[C1 && C2 && C3 && C4][[2 ;; 8]]},
 Flatten@X
]

It thus seems that the error is caused by a pathology in your constraints. Of course it may be that the pathology is inherit to your MWE. In that case proceeding by applying LogicalExpand to your complete constraints and trying to NMinimize with each toplevel Or statement may be useful to pin down the problem.

Update If the problems are indeed caused by incompatible linear equations in your real world problem than it might help to apply the following to you constraints cons:

LogicalExpand[cons] /. A : And[_Equal, (_Equal) ..] :> Reduce[A]

This will try to reduce the subpart of your constraints consisting of equalities, eliminating those branches of the expanded constraints with incompatible equalities. This works for your MWE at least.