• Gurobi Staff

Hi Sophia,

I am not sure if I understood it correctly but does Gurobi only make use of partial B&B or does it also apply then McCormicks envelope?

I am not sure what a partial B&B is. Gurobi uses the so-called spatial B&B algorithm where it branches on continuous variables and computes a valid relaxation of the model at each node of the B&B tree very similar to the classical discrete B&B algorithm. In order to construct the relaxation, Gurobi applies the McCormick envelope to every bilinear product or uses tangents/secants for quadratic terms.

I have implemented a non convex problem and then also the mccormick relaxation of this problem. Should I expect the same solution then?

Gurobi applies the McCormick envelopes at every node of the B&B tree. From what you are saying it looks like you applied the McCormick envelopes to your original model, i.e., before branching on any variable. This should be equivalent to the relaxation of the root node constructed by Gurobi if one would exclude Presolve.

Best regards,
Jaromił

Hello Jaromił Najman,

I found that the root relaxation excluding Presolve is apparently not the same as the relaxation that is obtained by just adding the McCormick envelopes.

I assume that Gurobi applies some domain reduction techniques (althoug Presolve=0) before adding the McCormick envelopes (and maybe more?) in order to construct the root relaxation.

In case I am right, is there a way to disable all of these domain reduction / constraint propagation techniques such that the gurobi root relaxation (without cuts and without presolve) is exactly the same as the relaxation obtained only by the McCormick envelopes?

Best regards,

Jan

• Gurobi Staff

Hi Jan,

Setting Presolve=0 should do it. There might be some trivial bound reductions that would still be applied, like variable fixing for example.

Best regards,
Jaromił

Hi Jaromil,

It would be interesting for me which of these trivial bound reductions are applied in my specific case.

In my instance, I have 15 x-variables, 15 y-variables and product variables z_ij = x_i * y_j, all are bounded by 0 and 1.

Besides the equations for the product variables there are the following constraints:

R0: x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7] + x[8] + x[9] + x[10]  + x[11] + x[12] + x[13] + x[14] + x[15] = 1R1: x[1] + x[2] + x[3] >= 0.09R2: x[4] + x[5] + x[6] >= 0.12R3: x[7] + x[8] + x[9] >= 0.14R4: x[10] + x[11] + x[12] >= 0.18R5: x[13] + x[14] + x[15] >= 0.12R6: x[1] + x[2] + x[3] <= 0.48R7: x[4] + x[5] + x[6] <= 0.21R8: x[7] + x[8] + x[9] <= 0.34R9: x[10] + x[11] + x[12] <= 0.48R10: x[13] + x[14] + x[15] <= 0.39

Can you tell me what the trivial bound reductions are in this case?

Best,

Jan

• Gurobi Staff

Hi Jan,

From the ones you listed, none should be used with Presolve=0. Do you still observe a difference with Presolve=0?

Best regards,
Jaromił

Yes. When I solve with Presolve=0 and NodeLimit=0, such that the optimization terminates with the root relaxation without cuts (is that right?), then I get an objective value 4.381421504525e+01, the McCormick relaxation yields 47.18940000000003

Best, Jan

• Gurobi Staff

You should look at the root node relaxation value which is printed directly before the B&B algorithm output. Could you please post the log output you get?

• Gurobi Staff

Interesting, could you try removing everything except variable bounds and the bilinear terms?

In this case the values are the same.Jaromił Najman

I have tested the same problem (with all constraints) in SCIP. There I explicitly disabled propagation and then the values are the same. It would be interesting whether there is an analogous possibility for gurobi.

• Gurobi Staff

In this case the values are the same.Jaromił Najman

So there has to be a linear constraint which is somehow exploited before constructing the McCormick envelope. Could you try to find out which one it is and post a minimal model where you see a difference in lower bound with Presolve=0. It is possible that some reduction is not properly turned off when Presolve=0 and it would be good to find out.

It turned out, that even only

R0: x[1] + x[2] + x[3] <= 0.48

yields a difference in the values.

• Gurobi Staff

Could you please construct a minimal model for which you can observe the difference and share it here. I would like to investigate it then.

Here is the .lp file

\ LP format - for model browsing. Use MPS format to capture full model detail.Maximize  1.29 x[1] + 3.91 x[2] - 1.24 x[3] - 0.68 x[4] - 2.74 x[5] - 2.09 x[6]   + 4.72 x[7] - 1.2 x[8] + 4.61 x[9] + 4.14 x[10] + 0.96 x[11] - 2.4 x[12]   + 4.81 x[13] - 0.04 x[14] - 0.85 x[15] + 1.29 y[1] + 3.91 y[2]   - 1.24 y[3] - 0.68 y[4] - 2.74 y[5] - 2.09 y[6] + 4.72 y[7] - 1.2 y[8]   + 4.61 y[9] + 4.14 y[10] + 0.96 y[11] - 2.4 y[12] + 4.81 y[13]   - 0.04 y[14] - 0.85 y[15] + 4.01 z[1,1] - 3.87 z[1,2] - 0.31 z[1,3]   - 2.53 z[1,4] + 0.44 z[1,5] + 0.74 z[1,6] - 4.87 z[1,7] - 2.83 z[1,8]   - 2.21 z[1,9] + 4.16 z[1,10] + 2.66 z[1,11] - 3.4 z[1,12] + 2.97 z[1,13]   - 3.61 z[1,14] + 1.17 z[1,15] - 3.73 z[2,1] - 4.98 z[2,2] + 3.71 z[2,3]   - 2.91 z[2,4] - 2.85 z[2,5] + 4.82 z[2,6] + 3.72 z[2,7] - 2.11 z[2,8]   + 4.61 z[2,9] + 0.39 z[2,10] + 1.78 z[2,11] - 2.95 z[2,12]   + 4.41 z[2,13] + 1.91 z[2,14] + 4.67 z[2,15] + 3.94 z[3,1] - 2.01 z[3,2]   - 1.39 z[3,3] - 3.34 z[3,4] - 3.54 z[3,5] - 4.35 z[3,6] - 1.99 z[3,7]   + 1.03 z[3,8] - 4.97 z[3,9] + 1.78 z[3,10] - 1.62 z[3,11] - 1.9 z[3,12]   + 3.19 z[3,13] - 0.19 z[3,14] - 1.84 z[3,15] - 0.19 z[4,1] + 2.05 z[4,2]   - 4.43 z[4,3] + 4.75 z[4,4] - 4.77 z[4,5] + 2.5 z[4,6] + 3.45 z[4,7]   - 4.82 z[4,8] + 2.88 z[4,9] - 1.34 z[4,10] + 0.79 z[4,11] - 4.91 z[4,12]   - 4.53 z[4,13] - 3.19 z[4,14] + 4.55 z[4,15] - 3.03 z[5,1] + 2.56 z[5,2]   + 4.3 z[5,3] + 4.42 z[5,4] - 1.56 z[5,5] - 1.45 z[5,6] + 0.25 z[5,7]   + 2.76 z[5,8] - 3.92 z[5,9] + 2.48 z[5,10] + 2.97 z[5,11] + 3.6 z[5,12]   - 4.63 z[5,13] + 4.46 z[5,14] - 4.09 z[5,15] - 1.59 z[6,1] + 1.11 z[6,2]   + 4.18 z[6,3] - 1.6 z[6,4] + 4.24 z[6,5] + 0.45 z[6,6] - 1.88 z[6,7]   - 1.83 z[6,8] - 3.23 z[6,9] - 4.22 z[6,10] - 3.51 z[6,11] + 1.89 z[6,12]   + 4.97 z[6,13] - 3.38 z[6,14] - 4.51 z[6,15] + 4.87 z[7,1] + 0.34 z[7,2]   - 0.94 z[7,3] - 2.63 z[7,4] + 0.94 z[7,5] + 3.26 z[7,6] - 0.44 z[7,7]   - 0.78 z[7,8] - 4.44 z[7,9] + 4.16 z[7,10] - 4.67 z[7,11] - 0.06 z[7,12]   + 3.38 z[7,13] - 3.69 z[7,14] + 2.32 z[7,15] + 4.5 z[8,1] + 1.3 z[8,2]   + 2.88 z[8,3] - 3.93 z[8,4] - 0.65 z[8,5] - 3.51 z[8,6] + 3.45 z[8,7]   - 2.05 z[8,8] - 0.47 z[8,9] + 4.99 z[8,10] + 3.52 z[8,11] + 4.76 z[8,12]   - 0.46 z[8,13] - 0.12 z[8,14] + 2.3 z[8,15] - 0.21 z[9,1] - 2.09 z[9,2]   - 0.96 z[9,3] - 3.53 z[9,4] - 1.23 z[9,5] + 4.88 z[9,6] + 4.6 z[9,7]   + 1.27 z[9,8] - 0.01 z[9,9] - 1.62 z[9,10] - 4.11 z[9,11] - 2.28 z[9,12]   + 2.82 z[9,13] + 3.67 z[9,14] - 1.39 z[9,15] + 2.86 z[10,1]   + 2.75 z[10,2] + 1.95 z[10,3] + 1.64 z[10,4] + 2.6 z[10,5]   - 1.37 z[10,6] + 2.04 z[10,7] - 2.19 z[10,8] - 0.14 z[10,9]   + 2.7 z[10,10] + 1.91 z[10,11] - 2.06 z[10,12] + 4.46 z[10,13]   + 1.5 z[10,14] + 0.81 z[10,15] - 4.88 z[11,1] + 0.47 z[11,2]   - 2.49 z[11,3] + 1.72 z[11,4] - 0.37 z[11,5] + 3.17 z[11,6]   + 1.47 z[11,7] + 2.98 z[11,8] - 1.52 z[11,9] + 1.44 z[11,10]   + 2.38 z[11,11] + 3.28 z[11,12] - 1.5 z[11,13] + 3.43 z[11,14]   + 3.7 z[11,15] + 1.88 z[12,1] + 4.76 z[12,2] + 4.57 z[12,3]   + 0.18 z[12,4] + 0.29 z[12,5] - 3.34 z[12,6] + 3.37 z[12,7]   + 4.37 z[12,8] - 0.23 z[12,9] + 1.91 z[12,10] + 2.2 z[12,11]   + 2.3 z[12,12] - 3.28 z[12,13] + 2.8 z[12,14] + 0.81 z[12,15]   + 1.66 z[13,1] - 0.79 z[13,2] + 1.24 z[13,3] + 2.75 z[13,4]   + 1.37 z[13,5] + 2.2 z[13,6] - 4.72 z[13,7] - 3.4 z[13,8] - 0.59 z[13,9]   + 1.5 z[13,10] - 2.81 z[13,11] + 1.86 z[13,12] + 1.31 z[13,13]   - 4.58 z[13,14] - 0.28 z[13,15] - 2.74 z[14,1] - 4.46 z[14,2]   - 3.66 z[14,3] - 1.83 z[14,4] - 3.18 z[14,5] - 3.07 z[14,6]   - 4.64 z[14,7] - 0.35 z[14,8] - 1.2 z[14,9] + 1.12 z[14,10]   + 0.9 z[14,11] - 2.62 z[14,12] + 4.03 z[14,13] - 4.99 z[14,14]   - 0.95 z[14,15] - 2.21 z[15,1] - 0.9 z[15,2] - 3.85 z[15,3]   + 3.31 z[15,4] - 1.26 z[15,5] - 4.64 z[15,6] + 1.14 z[15,7]   - 4.05 z[15,8] + 0.45 z[15,9] - 1.61 z[15,10] + 0.81 z[15,11]   + 4.58 z[15,12] + 3.19 z[15,13] - 0.81 z[15,14] + 3.13 z[15,15]Subject To R0: x[1] + x[2] + x[3] <= 0.48 qc0: - z[1,1] + [ x[1] * y[1] ] = 0 qc1: - z[1,2] + [ x[1] * y[2] ] = 0 qc2: - z[1,3] + [ x[1] * y[3] ] = 0 qc3: - z[1,4] + [ x[1] * y[4] ] = 0 qc4: - z[1,5] + [ x[1] * y[5] ] = 0 qc5: - z[1,6] + [ x[1] * y[6] ] = 0 qc6: - z[1,7] + [ x[1] * y[7] ] = 0 qc7: - z[1,8] + [ x[1] * y[8] ] = 0 qc8: - z[1,9] + [ x[1] * y[9] ] = 0 qc9: - z[1,10] + [ x[1] * y[10] ] = 0 qc10: - z[1,11] + [ x[1] * y[11] ] = 0 qc11: - z[1,12] + [ x[1] * y[12] ] = 0 qc12: - z[1,13] + [ x[1] * y[13] ] = 0 qc13: - z[1,14] + [ x[1] * y[14] ] = 0 qc14: - z[1,15] + [ x[1] * y[15] ] = 0 qc15: - z[2,1] + [ x[2] * y[1] ] = 0 qc16: - z[2,2] + [ x[2] * y[2] ] = 0 qc17: - z[2,3] + [ x[2] * y[3] ] = 0 qc18: - z[2,4] + [ x[2] * y[4] ] = 0 qc19: - z[2,5] + [ x[2] * y[5] ] = 0 qc20: - z[2,6] + [ x[2] * y[6] ] = 0 qc21: - z[2,7] + [ x[2] * y[7] ] = 0 qc22: - z[2,8] + [ x[2] * y[8] ] = 0 qc23: - z[2,9] + [ x[2] * y[9] ] = 0 qc24: - z[2,10] + [ x[2] * y[10] ] = 0 qc25: - z[2,11] + [ x[2] * y[11] ] = 0 qc26: - z[2,12] + [ x[2] * y[12] ] = 0 qc27: - z[2,13] + [ x[2] * y[13] ] = 0 qc28: - z[2,14] + [ x[2] * y[14] ] = 0 qc29: - z[2,15] + [ x[2] * y[15] ] = 0 qc30: - z[3,1] + [ x[3] * y[1] ] = 0 qc31: - z[3,2] + [ x[3] * y[2] ] = 0 qc32: - z[3,3] + [ x[3] * y[3] ] = 0 qc33: - z[3,4] + [ x[3] * y[4] ] = 0 qc34: - z[3,5] + [ x[3] * y[5] ] = 0 qc35: - z[3,6] + [ x[3] * y[6] ] = 0 qc36: - z[3,7] + [ x[3] * y[7] ] = 0 qc37: - z[3,8] + [ x[3] * y[8] ] = 0 qc38: - z[3,9] + [ x[3] * y[9] ] = 0 qc39: - z[3,10] + [ x[3] * y[10] ] = 0 qc40: - z[3,11] + [ x[3] * y[11] ] = 0 qc41: - z[3,12] + [ x[3] * y[12] ] = 0 qc42: - z[3,13] + [ x[3] * y[13] ] = 0 qc43: - z[3,14] + [ x[3] * y[14] ] = 0 qc44: - z[3,15] + [ x[3] * y[15] ] = 0 qc45: - z[4,1] + [ x[4] * y[1] ] = 0 qc46: - z[4,2] + [ x[4] * y[2] ] = 0 qc47: - z[4,3] + [ x[4] * y[3] ] = 0 qc48: - z[4,4] + [ x[4] * y[4] ] = 0 qc49: - z[4,5] + [ x[4] * y[5] ] = 0 qc50: - z[4,6] + [ x[4] * y[6] ] = 0 qc51: - z[4,7] + [ x[4] * y[7] ] = 0 qc52: - z[4,8] + [ x[4] * y[8] ] = 0 qc53: - z[4,9] + [ x[4] * y[9] ] = 0 qc54: - z[4,10] + [ x[4] * y[10] ] = 0 qc55: - z[4,11] + [ x[4] * y[11] ] = 0 qc56: - z[4,12] + [ x[4] * y[12] ] = 0 qc57: - z[4,13] + [ x[4] * y[13] ] = 0 qc58: - z[4,14] + [ x[4] * y[14] ] = 0 qc59: - z[4,15] + [ x[4] * y[15] ] = 0 qc60: - z[5,1] + [ x[5] * y[1] ] = 0 qc61: - z[5,2] + [ x[5] * y[2] ] = 0 qc62: - z[5,3] + [ x[5] * y[3] ] = 0 qc63: - z[5,4] + [ x[5] * y[4] ] = 0 qc64: - z[5,5] + [ x[5] * y[5] ] = 0 qc65: - z[5,6] + [ x[5] * y[6] ] = 0 qc66: - z[5,7] + [ x[5] * y[7] ] = 0 qc67: - z[5,8] + [ x[5] * y[8] ] = 0 qc68: - z[5,9] + [ x[5] * y[9] ] = 0 qc69: - z[5,10] + [ x[5] * y[10] ] = 0 qc70: - z[5,11] + [ x[5] * y[11] ] = 0 qc71: - z[5,12] + [ x[5] * y[12] ] = 0 qc72: - z[5,13] + [ x[5] * y[13] ] = 0 qc73: - z[5,14] + [ x[5] * y[14] ] = 0 qc74: - z[5,15] + [ x[5] * y[15] ] = 0 qc75: - z[6,1] + [ x[6] * y[1] ] = 0 qc76: - z[6,2] + [ x[6] * y[2] ] = 0 qc77: - z[6,3] + [ x[6] * y[3] ] = 0 qc78: - z[6,4] + [ x[6] * y[4] ] = 0 qc79: - z[6,5] + [ x[6] * y[5] ] = 0 qc80: - z[6,6] + [ x[6] * y[6] ] = 0 qc81: - z[6,7] + [ x[6] * y[7] ] = 0 qc82: - z[6,8] + [ x[6] * y[8] ] = 0 qc83: - z[6,9] + [ x[6] * y[9] ] = 0 qc84: - z[6,10] + [ x[6] * y[10] ] = 0 qc85: - z[6,11] + [ x[6] * y[11] ] = 0 qc86: - z[6,12] + [ x[6] * y[12] ] = 0 qc87: - z[6,13] + [ x[6] * y[13] ] = 0 qc88: - z[6,14] + [ x[6] * y[14] ] = 0 qc89: - z[6,15] + [ x[6] * y[15] ] = 0 qc90: - z[7,1] + [ x[7] * y[1] ] = 0 qc91: - z[7,2] + [ x[7] * y[2] ] = 0 qc92: - z[7,3] + [ x[7] * y[3] ] = 0 qc93: - z[7,4] + [ x[7] * y[4] ] = 0 qc94: - z[7,5] + [ x[7] * y[5] ] = 0 qc95: - z[7,6] + [ x[7] * y[6] ] = 0 qc96: - z[7,7] + [ x[7] * y[7] ] = 0 qc97: - z[7,8] + [ x[7] * y[8] ] = 0 qc98: - z[7,9] + [ x[7] * y[9] ] = 0 qc99: - z[7,10] + [ x[7] * y[10] ] = 0 qc100: - z[7,11] + [ x[7] * y[11] ] = 0 qc101: - z[7,12] + [ x[7] * y[12] ] = 0 qc102: - z[7,13] + [ x[7] * y[13] ] = 0 qc103: - z[7,14] + [ x[7] * y[14] ] = 0 qc104: - z[7,15] + [ x[7] * y[15] ] = 0 qc105: - z[8,1] + [ x[8] * y[1] ] = 0 qc106: - z[8,2] + [ x[8] * y[2] ] = 0 qc107: - z[8,3] + [ x[8] * y[3] ] = 0 qc108: - z[8,4] + [ x[8] * y[4] ] = 0 qc109: - z[8,5] + [ x[8] * y[5] ] = 0 qc110: - z[8,6] + [ x[8] * y[6] ] = 0 qc111: - z[8,7] + [ x[8] * y[7] ] = 0 qc112: - z[8,8] + [ x[8] * y[8] ] = 0 qc113: - z[8,9] + [ x[8] * y[9] ] = 0 qc114: - z[8,10] + [ x[8] * y[10] ] = 0 qc115: - z[8,11] + [ x[8] * y[11] ] = 0 qc116: - z[8,12] + [ x[8] * y[12] ] = 0 qc117: - z[8,13] + [ x[8] * y[13] ] = 0 qc118: - z[8,14] + [ x[8] * y[14] ] = 0 qc119: - z[8,15] + [ x[8] * y[15] ] = 0 qc120: - z[9,1] + [ x[9] * y[1] ] = 0 qc121: - z[9,2] + [ x[9] * y[2] ] = 0 qc122: - z[9,3] + [ x[9] * y[3] ] = 0 qc123: - z[9,4] + [ x[9] * y[4] ] = 0 qc124: - z[9,5] + [ x[9] * y[5] ] = 0 qc125: - z[9,6] + [ x[9] * y[6] ] = 0 qc126: - z[9,7] + [ x[9] * y[7] ] = 0 qc127: - z[9,8] + [ x[9] * y[8] ] = 0 qc128: - z[9,9] + [ x[9] * y[9] ] = 0 qc129: - z[9,10] + [ x[9] * y[10] ] = 0 qc130: - z[9,11] + [ x[9] * y[11] ] = 0 qc131: - z[9,12] + [ x[9] * y[12] ] = 0 qc132: - z[9,13] + [ x[9] * y[13] ] = 0 qc133: - z[9,14] + [ x[9] * y[14] ] = 0 qc134: - z[9,15] + [ x[9] * y[15] ] = 0 qc135: - z[10,1] + [ x[10] * y[1] ] = 0 qc136: - z[10,2] + [ x[10] * y[2] ] = 0 qc137: - z[10,3] + [ x[10] * y[3] ] = 0 qc138: - z[10,4] + [ x[10] * y[4] ] = 0 qc139: - z[10,5] + [ x[10] * y[5] ] = 0 qc140: - z[10,6] + [ x[10] * y[6] ] = 0 qc141: - z[10,7] + [ x[10] * y[7] ] = 0 qc142: - z[10,8] + [ x[10] * y[8] ] = 0 qc143: - z[10,9] + [ x[10] * y[9] ] = 0 qc144: - z[10,10] + [ x[10] * y[10] ] = 0 qc145: - z[10,11] + [ x[10] * y[11] ] = 0 qc146: - z[10,12] + [ x[10] * y[12] ] = 0 qc147: - z[10,13] + [ x[10] * y[13] ] = 0 qc148: - z[10,14] + [ x[10] * y[14] ] = 0 qc149: - z[10,15] + [ x[10] * y[15] ] = 0 qc150: - z[11,1] + [ x[11] * y[1] ] = 0 qc151: - z[11,2] + [ x[11] * y[2] ] = 0 qc152: - z[11,3] + [ x[11] * y[3] ] = 0 qc153: - z[11,4] + [ x[11] * y[4] ] = 0 qc154: - z[11,5] + [ x[11] * y[5] ] = 0 qc155: - z[11,6] + [ x[11] * y[6] ] = 0 qc156: - z[11,7] + [ x[11] * y[7] ] = 0 qc157: - z[11,8] + [ x[11] * y[8] ] = 0 qc158: - z[11,9] + [ x[11] * y[9] ] = 0 qc159: - z[11,10] + [ x[11] * y[10] ] = 0 qc160: - z[11,11] + [ x[11] * y[11] ] = 0 qc161: - z[11,12] + [ x[11] * y[12] ] = 0 qc162: - z[11,13] + [ x[11] * y[13] ] = 0 qc163: - z[11,14] + [ x[11] * y[14] ] = 0 qc164: - z[11,15] + [ x[11] * y[15] ] = 0 qc165: - z[12,1] + [ x[12] * y[1] ] = 0 qc166: - z[12,2] + [ x[12] * y[2] ] = 0 qc167: - z[12,3] + [ x[12] * y[3] ] = 0 qc168: - z[12,4] + [ x[12] * y[4] ] = 0 qc169: - z[12,5] + [ x[12] * y[5] ] = 0 qc170: - z[12,6] + [ x[12] * y[6] ] = 0 qc171: - z[12,7] + [ x[12] * y[7] ] = 0 qc172: - z[12,8] + [ x[12] * y[8] ] = 0 qc173: - z[12,9] + [ x[12] * y[9] ] = 0 qc174: - z[12,10] + [ x[12] * y[10] ] = 0 qc175: - z[12,11] + [ x[12] * y[11] ] = 0 qc176: - z[12,12] + [ x[12] * y[12] ] = 0 qc177: - z[12,13] + [ x[12] * y[13] ] = 0 qc178: - z[12,14] + [ x[12] * y[14] ] = 0 qc179: - z[12,15] + [ x[12] * y[15] ] = 0 qc180: - z[13,1] + [ x[13] * y[1] ] = 0 qc181: - z[13,2] + [ x[13] * y[2] ] = 0 qc182: - z[13,3] + [ x[13] * y[3] ] = 0 qc183: - z[13,4] + [ x[13] * y[4] ] = 0 qc184: - z[13,5] + [ x[13] * y[5] ] = 0 qc185: - z[13,6] + [ x[13] * y[6] ] = 0 qc186: - z[13,7] + [ x[13] * y[7] ] = 0 qc187: - z[13,8] + [ x[13] * y[8] ] = 0 qc188: - z[13,9] + [ x[13] * y[9] ] = 0 qc189: - z[13,10] + [ x[13] * y[10] ] = 0 qc190: - z[13,11] + [ x[13] * y[11] ] = 0 qc191: - z[13,12] + [ x[13] * y[12] ] = 0 qc192: - z[13,13] + [ x[13] * y[13] ] = 0 qc193: - z[13,14] + [ x[13] * y[14] ] = 0 qc194: - z[13,15] + [ x[13] * y[15] ] = 0 qc195: - z[14,1] + [ x[14] * y[1] ] = 0 qc196: - z[14,2] + [ x[14] * y[2] ] = 0 qc197: - z[14,3] + [ x[14] * y[3] ] = 0 qc198: - z[14,4] + [ x[14] * y[4] ] = 0 qc199: - z[14,5] + [ x[14] * y[5] ] = 0 qc200: - z[14,6] + [ x[14] * y[6] ] = 0 qc201: - z[14,7] + [ x[14] * y[7] ] = 0 qc202: - z[14,8] + [ x[14] * y[8] ] = 0 qc203: - z[14,9] + [ x[14] * y[9] ] = 0 qc204: - z[14,10] + [ x[14] * y[10] ] = 0 qc205: - z[14,11] + [ x[14] * y[11] ] = 0 qc206: - z[14,12] + [ x[14] * y[12] ] = 0 qc207: - z[14,13] + [ x[14] * y[13] ] = 0 qc208: - z[14,14] + [ x[14] * y[14] ] = 0 qc209: - z[14,15] + [ x[14] * y[15] ] = 0 qc210: - z[15,1] + [ x[15] * y[1] ] = 0 qc211: - z[15,2] + [ x[15] * y[2] ] = 0 qc212: - z[15,3] + [ x[15] * y[3] ] = 0 qc213: - z[15,4] + [ x[15] * y[4] ] = 0 qc214: - z[15,5] + [ x[15] * y[5] ] = 0 qc215: - z[15,6] + [ x[15] * y[6] ] = 0 qc216: - z[15,7] + [ x[15] * y[7] ] = 0 qc217: - z[15,8] + [ x[15] * y[8] ] = 0 qc218: - z[15,9] + [ x[15] * y[9] ] = 0 qc219: - z[15,10] + [ x[15] * y[10] ] = 0 qc220: - z[15,11] + [ x[15] * y[11] ] = 0 qc221: - z[15,12] + [ x[15] * y[12] ] = 0 qc222: - z[15,13] + [ x[15] * y[13] ] = 0 qc223: - z[15,14] + [ x[15] * y[14] ] = 0 qc224: - z[15,15] + [ x[15] * y[15] ] = 0Bounds x[1] <= 1 x[2] <= 1 x[3] <= 1 x[4] <= 1 x[5] <= 1 x[6] <= 1 x[7] <= 1 x[8] <= 1 x[9] <= 1 x[10] <= 1 x[11] <= 1 x[12] <= 1 x[13] <= 1 x[14] <= 1 x[15] <= 1 y[1] <= 1 y[2] <= 1 y[3] <= 1 y[4] <= 1 y[5] <= 1 y[6] <= 1 y[7] <= 1 y[8] <= 1 y[9] <= 1 y[10] <= 1 y[11] <= 1 y[12] <= 1 y[13] <= 1 y[14] <= 1 y[15] <= 1 z[1,1] <= 1 z[1,2] <= 1 z[1,3] <= 1 z[1,4] <= 1 z[1,5] <= 1 z[1,6] <= 1 z[1,7] <= 1 z[1,8] <= 1 z[1,9] <= 1 z[1,10] <= 1 z[1,11] <= 1 z[1,12] <= 1 z[1,13] <= 1 z[1,14] <= 1 z[1,15] <= 1 z[2,1] <= 1 z[2,2] <= 1 z[2,3] <= 1 z[2,4] <= 1 z[2,5] <= 1 z[2,6] <= 1 z[2,7] <= 1 z[2,8] <= 1 z[2,9] <= 1 z[2,10] <= 1 z[2,11] <= 1 z[2,12] <= 1 z[2,13] <= 1 z[2,14] <= 1 z[2,15] <= 1 z[3,1] <= 1 z[3,2] <= 1 z[3,3] <= 1 z[3,4] <= 1 z[3,5] <= 1 z[3,6] <= 1 z[3,7] <= 1 z[3,8] <= 1 z[3,9] <= 1 z[3,10] <= 1 z[3,11] <= 1 z[3,12] <= 1 z[3,13] <= 1 z[3,14] <= 1 z[3,15] <= 1 z[4,1] <= 1 z[4,2] <= 1 z[4,3] <= 1 z[4,4] <= 1 z[4,5] <= 1 z[4,6] <= 1 z[4,7] <= 1 z[4,8] <= 1 z[4,9] <= 1 z[4,10] <= 1 z[4,11] <= 1 z[4,12] <= 1 z[4,13] <= 1 z[4,14] <= 1 z[4,15] <= 1 z[5,1] <= 1 z[5,2] <= 1 z[5,3] <= 1 z[5,4] <= 1 z[5,5] <= 1 z[5,6] <= 1 z[5,7] <= 1 z[5,8] <= 1 z[5,9] <= 1 z[5,10] <= 1 z[5,11] <= 1 z[5,12] <= 1 z[5,13] <= 1 z[5,14] <= 1 z[5,15] <= 1 z[6,1] <= 1 z[6,2] <= 1 z[6,3] <= 1 z[6,4] <= 1 z[6,5] <= 1 z[6,6] <= 1 z[6,7] <= 1 z[6,8] <= 1 z[6,9] <= 1 z[6,10] <= 1 z[6,11] <= 1 z[6,12] <= 1 z[6,13] <= 1 z[6,14] <= 1 z[6,15] <= 1 z[7,1] <= 1 z[7,2] <= 1 z[7,3] <= 1 z[7,4] <= 1 z[7,5] <= 1 z[7,6] <= 1 z[7,7] <= 1 z[7,8] <= 1 z[7,9] <= 1 z[7,10] <= 1 z[7,11] <= 1 z[7,12] <= 1 z[7,13] <= 1 z[7,14] <= 1 z[7,15] <= 1 z[8,1] <= 1 z[8,2] <= 1 z[8,3] <= 1 z[8,4] <= 1 z[8,5] <= 1 z[8,6] <= 1 z[8,7] <= 1 z[8,8] <= 1 z[8,9] <= 1 z[8,10] <= 1 z[8,11] <= 1 z[8,12] <= 1 z[8,13] <= 1 z[8,14] <= 1 z[8,15] <= 1 z[9,1] <= 1 z[9,2] <= 1 z[9,3] <= 1 z[9,4] <= 1 z[9,5] <= 1 z[9,6] <= 1 z[9,7] <= 1 z[9,8] <= 1 z[9,9] <= 1 z[9,10] <= 1 z[9,11] <= 1 z[9,12] <= 1 z[9,13] <= 1 z[9,14] <= 1 z[9,15] <= 1 z[10,1] <= 1 z[10,2] <= 1 z[10,3] <= 1 z[10,4] <= 1 z[10,5] <= 1 z[10,6] <= 1 z[10,7] <= 1 z[10,8] <= 1 z[10,9] <= 1 z[10,10] <= 1 z[10,11] <= 1 z[10,12] <= 1 z[10,13] <= 1 z[10,14] <= 1 z[10,15] <= 1 z[11,1] <= 1 z[11,2] <= 1 z[11,3] <= 1 z[11,4] <= 1 z[11,5] <= 1 z[11,6] <= 1 z[11,7] <= 1 z[11,8] <= 1 z[11,9] <= 1 z[11,10] <= 1 z[11,11] <= 1 z[11,12] <= 1 z[11,13] <= 1 z[11,14] <= 1 z[11,15] <= 1 z[12,1] <= 1 z[12,2] <= 1 z[12,3] <= 1 z[12,4] <= 1 z[12,5] <= 1 z[12,6] <= 1 z[12,7] <= 1 z[12,8] <= 1 z[12,9] <= 1 z[12,10] <= 1 z[12,11] <= 1 z[12,12] <= 1 z[12,13] <= 1 z[12,14] <= 1 z[12,15] <= 1 z[13,1] <= 1 z[13,2] <= 1 z[13,3] <= 1 z[13,4] <= 1 z[13,5] <= 1 z[13,6] <= 1 z[13,7] <= 1 z[13,8] <= 1 z[13,9] <= 1 z[13,10] <= 1 z[13,11] <= 1 z[13,12] <= 1 z[13,13] <= 1 z[13,14] <= 1 z[13,15] <= 1 z[14,1] <= 1 z[14,2] <= 1 z[14,3] <= 1 z[14,4] <= 1 z[14,5] <= 1 z[14,6] <= 1 z[14,7] <= 1 z[14,8] <= 1 z[14,9] <= 1 z[14,10] <= 1 z[14,11] <= 1 z[14,12] <= 1 z[14,13] <= 1 z[14,14] <= 1 z[14,15] <= 1 z[15,1] <= 1 z[15,2] <= 1 z[15,3] <= 1 z[15,4] <= 1 z[15,5] <= 1 z[15,6] <= 1 z[15,7] <= 1 z[15,8] <= 1 z[15,9] <= 1 z[15,10] <= 1 z[15,11] <= 1 z[15,12] <= 1 z[15,13] <= 1 z[15,14] <= 1 z[15,15] <= 1End
• Gurobi Staff

Thank you, I'll try to find out something. Just to be sure, in your manual McCormick relaxation, you also added the constraint

R0: x[1] + x[2] + x[3] <= 0.48

correct?

Yes, that is correct.

• Gurobi Staff

Looking at the presolved model, which you can get via the presolve() method, I don't see any bound tightenings on any variable. I can also see all McCormick envelope constraints in the presolved model. Maybe you could check the presolved model against your manual construction? You can write the presolved model to a human readable file via

import gurobipy as gpm = gp.read("myLP.lp")m.setParam("NonConvex",2)m.setParam("Presolve",0)p = m.presolve()p.write("presolve.lp")

You can open the file $$\texttt{presolve.lp}$$ in any standard text editor.

If you leave Presolve turned on, i.e., you remove the line

m.setParam("Presolve",0)

you will see in the presolved model that Gurobi improved some bounds using the R0 inequality.

It seems that

- the presolved model (having Presolve=0) has R0+all Mccormick envelopes (explicitly) + all product variable equations + trivial bounds 0 and 1

- my manual McCormick relaxation has R0 + all Mccormick envelopes but z_i,j >= 0 (it is implicitly given with the bounds) + trivial bounds 0 and 1

So when one deletes the bilinear equations in the presolved model (with presolve=0) one exactly arrives at the manual McCormick relaxation. But this would mean, that Gurobi makes some different changes, that are independent of presolve, in order to compute the root relaxation, right?

Here is the .lp file for the presolved model with presolve=0:

Maximize  1.29 x[1] + 3.91 x[2] - 1.24 x[3] - 0.68 x[4] - 2.74 x[5] - 2.09 x[6]   + 4.72 x[7] - 1.2 x[8] + 4.61 x[9] + 4.14 x[10] + 0.96 x[11] - 2.4 x[12]   + 4.81 x[13] - 0.04 x[14] - 0.85 x[15] + 1.29 y[1] + 3.91 y[2]   - 1.24 y[3] - 0.68 y[4] - 2.74 y[5] - 2.09 y[6] + 4.72 y[7] - 1.2 y[8]   + 4.61 y[9] + 4.14 y[10] + 0.96 y[11] - 2.4 y[12] + 4.81 y[13]   - 0.04 y[14] - 0.85 y[15] + 4.01 z[1,1] - 3.87 z[1,2] - 0.31 z[1,3]   - 2.53 z[1,4] + 0.44 z[1,5] + 0.74 z[1,6] - 4.87 z[1,7] - 2.83 z[1,8]   - 2.21 z[1,9] + 4.16 z[1,10] + 2.66 z[1,11] - 3.4 z[1,12] + 2.97 z[1,13]   - 3.61 z[1,14] + 1.17 z[1,15] - 3.73 z[2,1] - 4.98 z[2,2] + 3.71 z[2,3]   - 2.91 z[2,4] - 2.85 z[2,5] + 4.82 z[2,6] + 3.72 z[2,7] - 2.11 z[2,8]   + 4.61 z[2,9] + 0.39 z[2,10] + 1.78 z[2,11] - 2.95 z[2,12]   + 4.41 z[2,13] + 1.91 z[2,14] + 4.67 z[2,15] + 3.94 z[3,1] - 2.01 z[3,2]   - 1.39 z[3,3] - 3.34 z[3,4] - 3.54 z[3,5] - 4.35 z[3,6] - 1.99 z[3,7]   + 1.03 z[3,8] - 4.97 z[3,9] + 1.78 z[3,10] - 1.62 z[3,11] - 1.9 z[3,12]   + 3.19 z[3,13] - 0.19 z[3,14] - 1.84 z[3,15] - 0.19 z[4,1] + 2.05 z[4,2]   - 4.43 z[4,3] + 4.75 z[4,4] - 4.77 z[4,5] + 2.5 z[4,6] + 3.45 z[4,7]   - 4.82 z[4,8] + 2.88 z[4,9] - 1.34 z[4,10] + 0.79 z[4,11] - 4.91 z[4,12]   - 4.53 z[4,13] - 3.19 z[4,14] + 4.55 z[4,15] - 3.03 z[5,1] + 2.56 z[5,2]   + 4.3 z[5,3] + 4.42 z[5,4] - 1.56 z[5,5] - 1.45 z[5,6] + 0.25 z[5,7]   + 2.76 z[5,8] - 3.92 z[5,9] + 2.48 z[5,10] + 2.97 z[5,11] + 3.6 z[5,12]   - 4.63 z[5,13] + 4.46 z[5,14] - 4.09 z[5,15] - 1.59 z[6,1] + 1.11 z[6,2]   + 4.18 z[6,3] - 1.6 z[6,4] + 4.24 z[6,5] + 0.45 z[6,6] - 1.88 z[6,7]   - 1.83 z[6,8] - 3.23 z[6,9] - 4.22 z[6,10] - 3.51 z[6,11] + 1.89 z[6,12]   + 4.97 z[6,13] - 3.38 z[6,14] - 4.51 z[6,15] + 4.87 z[7,1] + 0.34 z[7,2]   - 0.94 z[7,3] - 2.63 z[7,4] + 0.94 z[7,5] + 3.26 z[7,6] - 0.44 z[7,7]   - 0.78 z[7,8] - 4.44 z[7,9] + 4.16 z[7,10] - 4.67 z[7,11] - 0.06 z[7,12]   + 3.38 z[7,13] - 3.69 z[7,14] + 2.32 z[7,15] + 4.5 z[8,1] + 1.3 z[8,2]   + 2.88 z[8,3] - 3.93 z[8,4] - 0.65 z[8,5] - 3.51 z[8,6] + 3.45 z[8,7]   - 2.05 z[8,8] - 0.47 z[8,9] + 4.99 z[8,10] + 3.52 z[8,11] + 4.76 z[8,12]   - 0.46 z[8,13] - 0.12 z[8,14] + 2.3 z[8,15] - 0.21 z[9,1] - 2.09 z[9,2]   - 0.96 z[9,3] - 3.53 z[9,4] - 1.23 z[9,5] + 4.88 z[9,6] + 4.6 z[9,7]   + 1.27 z[9,8] - 0.01 z[9,9] - 1.62 z[9,10] - 4.11 z[9,11] - 2.28 z[9,12]   + 2.82 z[9,13] + 3.67 z[9,14] - 1.39 z[9,15] + 2.86 z[10,1]   + 2.75 z[10,2] + 1.95 z[10,3] + 1.64 z[10,4] + 2.6 z[10,5]   - 1.37 z[10,6] + 2.04 z[10,7] - 2.19 z[10,8] - 0.14 z[10,9]   + 2.7 z[10,10] + 1.91 z[10,11] - 2.06 z[10,12] + 4.46 z[10,13]   + 1.5 z[10,14] + 0.81 z[10,15] - 4.88 z[11,1] + 0.47 z[11,2]   - 2.49 z[11,3] + 1.72 z[11,4] - 0.37 z[11,5] + 3.17 z[11,6]   + 1.47 z[11,7] + 2.98 z[11,8] - 1.52 z[11,9] + 1.44 z[11,10]   + 2.38 z[11,11] + 3.28 z[11,12] - 1.5 z[11,13] + 3.43 z[11,14]   + 3.7 z[11,15] + 1.88 z[12,1] + 4.76 z[12,2] + 4.57 z[12,3]   + 0.18 z[12,4] + 0.29 z[12,5] - 3.34 z[12,6] + 3.37 z[12,7]   + 4.37 z[12,8] - 0.23 z[12,9] + 1.91 z[12,10] + 2.2 z[12,11]   + 2.3 z[12,12] - 3.28 z[12,13] + 2.8 z[12,14] + 0.81 z[12,15]   + 1.66 z[13,1] - 0.79 z[13,2] + 1.24 z[13,3] + 2.75 z[13,4]   + 1.37 z[13,5] + 2.2 z[13,6] - 4.72 z[13,7] - 3.4 z[13,8] - 0.59 z[13,9]   + 1.5 z[13,10] - 2.81 z[13,11] + 1.86 z[13,12] + 1.31 z[13,13]   - 4.58 z[13,14] - 0.28 z[13,15] - 2.74 z[14,1] - 4.46 z[14,2]   - 3.66 z[14,3] - 1.83 z[14,4] - 3.18 z[14,5] - 3.07 z[14,6]   - 4.64 z[14,7] - 0.35 z[14,8] - 1.2 z[14,9] + 1.12 z[14,10]   + 0.9 z[14,11] - 2.62 z[14,12] + 4.03 z[14,13] - 4.99 z[14,14]   - 0.95 z[14,15] - 2.21 z[15,1] - 0.9 z[15,2] - 3.85 z[15,3]   + 3.31 z[15,4] - 1.26 z[15,5] - 4.64 z[15,6] + 1.14 z[15,7]   - 4.05 z[15,8] + 0.45 z[15,9] - 1.61 z[15,10] + 0.81 z[15,11]   + 4.58 z[15,12] + 3.19 z[15,13] - 0.81 z[15,14] + 3.13 z[15,15]Subject To R0: x[1] + x[2] + x[3] <= 0.48 R1: - y[1] + z[1,1] <= 0 R2: - x[1] + z[1,1] <= 0 R3: - z[1,1] <= 0 R4: x[1] + y[1] - z[1,1] <= 1 R5: - y[2] + z[1,2] <= 0 R6: - x[1] + z[1,2] <= 0 R7: - z[1,2] <= 0 R8: x[1] + y[2] - z[1,2] <= 1 R9: - y[3] + z[1,3] <= 0 R10: - x[1] + z[1,3] <= 0 R11: - z[1,3] <= 0 R12: x[1] + y[3] - z[1,3] <= 1 R13: - y[4] + z[1,4] <= 0 R14: - x[1] + z[1,4] <= 0 R15: - z[1,4] <= 0 R16: x[1] + y[4] - z[1,4] <= 1 R17: - y[5] + z[1,5] <= 0 R18: - x[1] + z[1,5] <= 0 R19: - z[1,5] <= 0 R20: x[1] + y[5] - z[1,5] <= 1 R21: - y[6] + z[1,6] <= 0 R22: - x[1] + z[1,6] <= 0 R23: - z[1,6] <= 0 R24: x[1] + y[6] - z[1,6] <= 1 R25: - y[7] + z[1,7] <= 0 R26: - x[1] + z[1,7] <= 0 R27: - z[1,7] <= 0 R28: x[1] + y[7] - z[1,7] <= 1 R29: - y[8] + z[1,8] <= 0 R30: - x[1] + z[1,8] <= 0 R31: - z[1,8] <= 0 R32: x[1] + y[8] - z[1,8] <= 1 R33: - y[9] + z[1,9] <= 0 R34: - x[1] + z[1,9] <= 0 R35: - z[1,9] <= 0 R36: x[1] + y[9] - z[1,9] <= 1 R37: - y[10] + z[1,10] <= 0 R38: - x[1] + z[1,10] <= 0 R39: - z[1,10] <= 0 R40: x[1] + y[10] - z[1,10] <= 1 R41: - y[11] + z[1,11] <= 0 R42: - x[1] + z[1,11] <= 0 R43: - z[1,11] <= 0 R44: x[1] + y[11] - z[1,11] <= 1 R45: - y[12] + z[1,12] <= 0 R46: - x[1] + z[1,12] <= 0 R47: - z[1,12] <= 0 R48: x[1] + y[12] - z[1,12] <= 1 R49: - y[13] + z[1,13] <= 0 R50: - x[1] + z[1,13] <= 0 R51: - z[1,13] <= 0 R52: x[1] + y[13] - z[1,13] <= 1 R53: - y[14] + z[1,14] <= 0 R54: - x[1] + z[1,14] <= 0 R55: - z[1,14] <= 0 R56: x[1] + y[14] - z[1,14] <= 1 R57: - y[15] + z[1,15] <= 0 R58: - x[1] + z[1,15] <= 0 R59: - z[1,15] <= 0 R60: x[1] + y[15] - z[1,15] <= 1 R61: - y[1] + z[2,1] <= 0 R62: - x[2] + z[2,1] <= 0 R63: - z[2,1] <= 0 R64: x[2] + y[1] - z[2,1] <= 1 R65: - y[2] + z[2,2] <= 0 R66: - x[2] + z[2,2] <= 0 R67: - z[2,2] <= 0 R68: x[2] + y[2] - z[2,2] <= 1 R69: - y[3] + z[2,3] <= 0 R70: - x[2] + z[2,3] <= 0 R71: - z[2,3] <= 0 R72: x[2] + y[3] - z[2,3] <= 1 R73: - y[4] + z[2,4] <= 0 R74: - x[2] + z[2,4] <= 0 R75: - z[2,4] <= 0 R76: x[2] + y[4] - z[2,4] <= 1 R77: - y[5] + z[2,5] <= 0 R78: - x[2] + z[2,5] <= 0 R79: - z[2,5] <= 0 R80: x[2] + y[5] - z[2,5] <= 1 R81: - y[6] + z[2,6] <= 0 R82: - x[2] + z[2,6] <= 0 R83: - z[2,6] <= 0 R84: x[2] + y[6] - z[2,6] <= 1 R85: - y[7] + z[2,7] <= 0 R86: - x[2] + z[2,7] <= 0 R87: - z[2,7] <= 0 R88: x[2] + y[7] - z[2,7] <= 1 R89: - y[8] + z[2,8] <= 0 R90: - x[2] + z[2,8] <= 0 R91: - z[2,8] <= 0 R92: x[2] + y[8] - z[2,8] <= 1 R93: - y[9] + z[2,9] <= 0 R94: - x[2] + z[2,9] <= 0 R95: - z[2,9] <= 0 R96: x[2] + y[9] - z[2,9] <= 1 R97: - y[10] + z[2,10] <= 0 R98: - x[2] + z[2,10] <= 0 R99: - z[2,10] <= 0 R100: x[2] + y[10] - z[2,10] <= 1 R101: - y[11] + z[2,11] <= 0 R102: - x[2] + z[2,11] <= 0 R103: - z[2,11] <= 0 R104: x[2] + y[11] - z[2,11] <= 1 R105: - y[12] + z[2,12] <= 0 R106: - x[2] + z[2,12] <= 0 R107: - z[2,12] <= 0 R108: x[2] + y[12] - z[2,12] <= 1 R109: - y[13] + z[2,13] <= 0 R110: - x[2] + z[2,13] <= 0 R111: - z[2,13] <= 0 R112: x[2] + y[13] - z[2,13] <= 1 R113: - y[14] + z[2,14] <= 0 R114: - x[2] + z[2,14] <= 0 R115: - z[2,14] <= 0 R116: x[2] + y[14] - z[2,14] <= 1 R117: - y[15] + z[2,15] <= 0 R118: - x[2] + z[2,15] <= 0 R119: - z[2,15] <= 0 R120: x[2] + y[15] - z[2,15] <= 1 R121: - y[1] + z[3,1] <= 0 R122: - x[3] + z[3,1] <= 0 R123: - z[3,1] <= 0 R124: x[3] + y[1] - z[3,1] <= 1 R125: - y[2] + z[3,2] <= 0 R126: - x[3] + z[3,2] <= 0 R127: - z[3,2] <= 0 R128: x[3] + y[2] - z[3,2] <= 1 R129: - y[3] + z[3,3] <= 0 R130: - x[3] + z[3,3] <= 0 R131: - z[3,3] <= 0 R132: x[3] + y[3] - z[3,3] <= 1 R133: - y[4] + z[3,4] <= 0 R134: - x[3] + z[3,4] <= 0 R135: - z[3,4] <= 0 R136: x[3] + y[4] - z[3,4] <= 1 R137: - y[5] + z[3,5] <= 0 R138: - x[3] + z[3,5] <= 0 R139: - z[3,5] <= 0 R140: x[3] + y[5] - z[3,5] <= 1 R141: - y[6] + z[3,6] <= 0 R142: - x[3] + z[3,6] <= 0 R143: - z[3,6] <= 0 R144: x[3] + y[6] - z[3,6] <= 1 R145: - y[7] + z[3,7] <= 0 R146: - x[3] + z[3,7] <= 0 R147: - z[3,7] <= 0 R148: x[3] + y[7] - z[3,7] <= 1 R149: - y[8] + z[3,8] <= 0 R150: - x[3] + z[3,8] <= 0 R151: - z[3,8] <= 0 R152: x[3] + y[8] - z[3,8] <= 1 R153: - y[9] + z[3,9] <= 0 R154: - x[3] + z[3,9] <= 0 R155: - z[3,9] <= 0 R156: x[3] + y[9] - z[3,9] <= 1 R157: - y[10] + z[3,10] <= 0 R158: - x[3] + z[3,10] <= 0 R159: - z[3,10] <= 0 R160: x[3] + y[10] - z[3,10] <= 1 R161: - y[11] + z[3,11] <= 0 R162: - x[3] + z[3,11] <= 0 R163: - z[3,11] <= 0 R164: x[3] + y[11] - z[3,11] <= 1 R165: - y[12] + z[3,12] <= 0 R166: - x[3] + z[3,12] <= 0 R167: - z[3,12] <= 0 R168: x[3] + y[12] - z[3,12] <= 1 R169: - y[13] + z[3,13] <= 0 R170: - x[3] + z[3,13] <= 0 R171: - z[3,13] <= 0 R172: x[3] + y[13] - z[3,13] <= 1 R173: - y[14] + z[3,14] <= 0 R174: - x[3] + z[3,14] <= 0 R175: - z[3,14] <= 0 R176: x[3] + y[14] - z[3,14] <= 1 R177: - y[15] + z[3,15] <= 0 R178: - x[3] + z[3,15] <= 0 R179: - z[3,15] <= 0 R180: x[3] + y[15] - z[3,15] <= 1 R181: - y[1] + z[4,1] <= 0 R182: - x[4] + z[4,1] <= 0 R183: - z[4,1] <= 0 R184: x[4] + y[1] - z[4,1] <= 1 R185: - y[2] + z[4,2] <= 0 R186: - x[4] + z[4,2] <= 0 R187: - z[4,2] <= 0 R188: x[4] + y[2] - z[4,2] <= 1 R189: - y[3] + z[4,3] <= 0 R190: - x[4] + z[4,3] <= 0 R191: - z[4,3] <= 0 R192: x[4] + y[3] - z[4,3] <= 1 R193: - y[4] + z[4,4] <= 0 R194: - x[4] + z[4,4] <= 0 R195: - z[4,4] <= 0 R196: x[4] + y[4] - z[4,4] <= 1 R197: - y[5] + z[4,5] <= 0 R198: - x[4] + z[4,5] <= 0 R199: - z[4,5] <= 0 R200: x[4] + y[5] - z[4,5] <= 1 R201: - y[6] + z[4,6] <= 0 R202: - x[4] + z[4,6] <= 0 R203: - z[4,6] <= 0 R204: x[4] + y[6] - z[4,6] <= 1 R205: - y[7] + z[4,7] <= 0 R206: - x[4] + z[4,7] <= 0 R207: - z[4,7] <= 0 R208: x[4] + y[7] - z[4,7] <= 1 R209: - y[8] + z[4,8] <= 0 R210: - x[4] + z[4,8] <= 0 R211: - z[4,8] <= 0 R212: x[4] + y[8] - z[4,8] <= 1 R213: - y[9] + z[4,9] <= 0 R214: - x[4] + z[4,9] <= 0 R215: - z[4,9] <= 0 R216: x[4] + y[9] - z[4,9] <= 1 R217: - y[10] + z[4,10] <= 0 R218: - x[4] + z[4,10] <= 0 R219: - z[4,10] <= 0 R220: x[4] + y[10] - z[4,10] <= 1 R221: - y[11] + z[4,11] <= 0 R222: - x[4] + z[4,11] <= 0 R223: - z[4,11] <= 0 R224: x[4] + y[11] - z[4,11] <= 1 R225: - y[12] + z[4,12] <= 0 R226: - x[4] + z[4,12] <= 0 R227: - z[4,12] <= 0 R228: x[4] + y[12] - z[4,12] <= 1 R229: - y[13] + z[4,13] <= 0 R230: - x[4] + z[4,13] <= 0 R231: - z[4,13] <= 0 R232: x[4] + y[13] - z[4,13] <= 1 R233: - y[14] + z[4,14] <= 0 R234: - x[4] + z[4,14] <= 0 R235: - z[4,14] <= 0 R236: x[4] + y[14] - z[4,14] <= 1 R237: - y[15] + z[4,15] <= 0 R238: - x[4] + z[4,15] <= 0 R239: - z[4,15] <= 0 R240: x[4] + y[15] - z[4,15] <= 1 R241: - y[1] + z[5,1] <= 0 R242: - x[5] + z[5,1] <= 0 R243: - z[5,1] <= 0 R244: x[5] + y[1] - z[5,1] <= 1 R245: - y[2] + z[5,2] <= 0 R246: - x[5] + z[5,2] <= 0 R247: - z[5,2] <= 0 R248: x[5] + y[2] - z[5,2] <= 1 R249: - y[3] + z[5,3] <= 0 R250: - x[5] + z[5,3] <= 0 R251: - z[5,3] <= 0 R252: x[5] + y[3] - z[5,3] <= 1 R253: - y[4] + z[5,4] <= 0 R254: - x[5] + z[5,4] <= 0 R255: - z[5,4] <= 0 R256: x[5] + y[4] - z[5,4] <= 1 R257: - y[5] + z[5,5] <= 0 R258: - x[5] + z[5,5] <= 0 R259: - z[5,5] <= 0 R260: x[5] + y[5] - z[5,5] <= 1 R261: - y[6] + z[5,6] <= 0 R262: - x[5] + z[5,6] <= 0 R263: - z[5,6] <= 0 R264: x[5] + y[6] - z[5,6] <= 1 R265: - y[7] + z[5,7] <= 0 R266: - x[5] + z[5,7] <= 0 R267: - z[5,7] <= 0 R268: x[5] + y[7] - z[5,7] <= 1 R269: - y[8] + z[5,8] <= 0 R270: - x[5] + z[5,8] <= 0 R271: - z[5,8] <= 0 R272: x[5] + y[8] - z[5,8] <= 1 R273: - y[9] + z[5,9] <= 0 R274: - x[5] + z[5,9] <= 0 R275: - z[5,9] <= 0 R276: x[5] + y[9] - z[5,9] <= 1 R277: - y[10] + z[5,10] <= 0 R278: - x[5] + z[5,10] <= 0 R279: - z[5,10] <= 0 R280: x[5] + y[10] - z[5,10] <= 1 R281: - y[11] + z[5,11] <= 0 R282: - x[5] + z[5,11] <= 0 R283: - z[5,11] <= 0 R284: x[5] + y[11] - z[5,11] <= 1 R285: - y[12] + z[5,12] <= 0 R286: - x[5] + z[5,12] <= 0 R287: - z[5,12] <= 0 R288: x[5] + y[12] - z[5,12] <= 1 R289: - y[13] + z[5,13] <= 0 R290: - x[5] + z[5,13] <= 0 R291: - z[5,13] <= 0 R292: x[5] + y[13] - z[5,13] <= 1 R293: - y[14] + z[5,14] <= 0 R294: - x[5] + z[5,14] <= 0 R295: - z[5,14] <= 0 R296: x[5] + y[14] - z[5,14] <= 1 R297: - y[15] + z[5,15] <= 0 R298: - x[5] + z[5,15] <= 0 R299: - z[5,15] <= 0 R300: x[5] + y[15] - z[5,15] <= 1 R301: - y[1] + z[6,1] <= 0 R302: - x[6] + z[6,1] <= 0 R303: - z[6,1] <= 0 R304: x[6] + y[1] - z[6,1] <= 1 R305: - y[2] + z[6,2] <= 0 R306: - x[6] + z[6,2] <= 0 R307: - z[6,2] <= 0 R308: x[6] + y[2] - z[6,2] <= 1 R309: - y[3] + z[6,3] <= 0 R310: - x[6] + z[6,3] <= 0 R311: - z[6,3] <= 0 R312: x[6] + y[3] - z[6,3] <= 1 R313: - y[4] + z[6,4] <= 0 R314: - x[6] + z[6,4] <= 0 R315: - z[6,4] <= 0 R316: x[6] + y[4] - z[6,4] <= 1 R317: - y[5] + z[6,5] <= 0 R318: - x[6] + z[6,5] <= 0 R319: - z[6,5] <= 0 R320: x[6] + y[5] - z[6,5] <= 1 R321: - y[6] + z[6,6] <= 0 R322: - x[6] + z[6,6] <= 0 R323: - z[6,6] <= 0 R324: x[6] + y[6] - z[6,6] <= 1 R325: - y[7] + z[6,7] <= 0 R326: - x[6] + z[6,7] <= 0 R327: - z[6,7] <= 0 R328: x[6] + y[7] - z[6,7] <= 1 R329: - y[8] + z[6,8] <= 0 R330: - x[6] + z[6,8] <= 0 R331: - z[6,8] <= 0 R332: x[6] + y[8] - z[6,8] <= 1 R333: - y[9] + z[6,9] <= 0 R334: - x[6] + z[6,9] <= 0 R335: - z[6,9] <= 0 R336: x[6] + y[9] - z[6,9] <= 1 R337: - y[10] + z[6,10] <= 0 R338: - x[6] + z[6,10] <= 0 R339: - z[6,10] <= 0 R340: x[6] + y[10] - z[6,10] <= 1 R341: - y[11] + z[6,11] <= 0 R342: - x[6] + z[6,11] <= 0 R343: - z[6,11] <= 0 R344: x[6] + y[11] - z[6,11] <= 1 R345: - y[12] + z[6,12] <= 0 R346: - x[6] + z[6,12] <= 0 R347: - z[6,12] <= 0 R348: x[6] + y[12] - z[6,12] <= 1 R349: - y[13] + z[6,13] <= 0 R350: - x[6] + z[6,13] <= 0 R351: - z[6,13] <= 0 R352: x[6] + y[13] - z[6,13] <= 1 R353: - y[14] + z[6,14] <= 0 R354: - x[6] + z[6,14] <= 0 R355: - z[6,14] <= 0 R356: x[6] + y[14] - z[6,14] <= 1 R357: - y[15] + z[6,15] <= 0 R358: - x[6] + z[6,15] <= 0 R359: - z[6,15] <= 0 R360: x[6] + y[15] - z[6,15] <= 1 R361: - y[1] + z[7,1] <= 0 R362: - x[7] + z[7,1] <= 0 R363: - z[7,1] <= 0 R364: x[7] + y[1] - z[7,1] <= 1 R365: - y[2] + z[7,2] <= 0 R366: - x[7] + z[7,2] <= 0 R367: - z[7,2] <= 0 R368: x[7] + y[2] - z[7,2] <= 1 R369: - y[3] + z[7,3] <= 0 R370: - x[7] + z[7,3] <= 0 R371: - z[7,3] <= 0 R372: x[7] + y[3] - z[7,3] <= 1 R373: - y[4] + z[7,4] <= 0 R374: - x[7] + z[7,4] <= 0 R375: - z[7,4] <= 0 R376: x[7] + y[4] - z[7,4] <= 1 R377: - y[5] + z[7,5] <= 0 R378: - x[7] + z[7,5] <= 0 R379: - z[7,5] <= 0 R380: x[7] + y[5] - z[7,5] <= 1 R381: - y[6] + z[7,6] <= 0 R382: - x[7] + z[7,6] <= 0 R383: - z[7,6] <= 0 R384: x[7] + y[6] - z[7,6] <= 1 R385: - y[7] + z[7,7] <= 0 R386: - x[7] + z[7,7] <= 0 R387: - z[7,7] <= 0 R388: x[7] + y[7] - z[7,7] <= 1 R389: - y[8] + z[7,8] <= 0 R390: - x[7] + z[7,8] <= 0 R391: - z[7,8] <= 0 R392: x[7] + y[8] - z[7,8] <= 1 R393: - y[9] + z[7,9] <= 0 R394: - x[7] + z[7,9] <= 0 R395: - z[7,9] <= 0 R396: x[7] + y[9] - z[7,9] <= 1 R397: - y[10] + z[7,10] <= 0 R398: - x[7] + z[7,10] <= 0 R399: - z[7,10] <= 0 R400: x[7] + y[10] - z[7,10] <= 1 R401: - y[11] + z[7,11] <= 0 R402: - x[7] + z[7,11] <= 0 R403: - z[7,11] <= 0 R404: x[7] + y[11] - z[7,11] <= 1 R405: - y[12] + z[7,12] <= 0 R406: - x[7] + z[7,12] <= 0 R407: - z[7,12] <= 0 R408: x[7] + y[12] - z[7,12] <= 1 R409: - y[13] + z[7,13] <= 0 R410: - x[7] + z[7,13] <= 0 R411: - z[7,13] <= 0 R412: x[7] + y[13] - z[7,13] <= 1 R413: - y[14] + z[7,14] <= 0 R414: - x[7] + z[7,14] <= 0 R415: - z[7,14] <= 0 R416: x[7] + y[14] - z[7,14] <= 1 R417: - y[15] + z[7,15] <= 0 R418: - x[7] + z[7,15] <= 0 R419: - z[7,15] <= 0 R420: x[7] + y[15] - z[7,15] <= 1 R421: - y[1] + z[8,1] <= 0 R422: - x[8] + z[8,1] <= 0 R423: - z[8,1] <= 0 R424: x[8] + y[1] - z[8,1] <= 1 R425: - y[2] + z[8,2] <= 0 R426: - x[8] + z[8,2] <= 0 R427: - z[8,2] <= 0 R428: x[8] + y[2] - z[8,2] <= 1 R429: - y[3] + z[8,3] <= 0 R430: - x[8] + z[8,3] <= 0 R431: - z[8,3] <= 0 R432: x[8] + y[3] - z[8,3] <= 1 R433: - y[4] + z[8,4] <= 0 R434: - x[8] + z[8,4] <= 0 R435: - z[8,4] <= 0 R436: x[8] + y[4] - z[8,4] <= 1 R437: - y[5] + z[8,5] <= 0 R438: - x[8] + z[8,5] <= 0 R439: - z[8,5] <= 0 R440: x[8] + y[5] - z[8,5] <= 1 R441: - y[6] + z[8,6] <= 0 R442: - x[8] + z[8,6] <= 0 R443: - z[8,6] <= 0 R444: x[8] + y[6] - z[8,6] <= 1 R445: - y[7] + z[8,7] <= 0 R446: - x[8] + z[8,7] <= 0 R447: - z[8,7] <= 0 R448: x[8] + y[7] - z[8,7] <= 1 R449: - y[8] + z[8,8] <= 0 R450: - x[8] + z[8,8] <= 0 R451: - z[8,8] <= 0 R452: x[8] + y[8] - z[8,8] <= 1 R453: - y[9] + z[8,9] <= 0 R454: - x[8] + z[8,9] <= 0 R455: - z[8,9] <= 0 R456: x[8] + y[9] - z[8,9] <= 1 R457: - y[10] + z[8,10] <= 0 R458: - x[8] + z[8,10] <= 0 R459: - z[8,10] <= 0 R460: x[8] + y[10] - z[8,10] <= 1 R461: - y[11] + z[8,11] <= 0 R462: - x[8] + z[8,11] <= 0 R463: - z[8,11] <= 0 R464: x[8] + y[11] - z[8,11] <= 1 R465: - y[12] + z[8,12] <= 0 R466: - x[8] + z[8,12] <= 0 R467: - z[8,12] <= 0 R468: x[8] + y[12] - z[8,12] <= 1 R469: - y[13] + z[8,13] <= 0 R470: - x[8] + z[8,13] <= 0 R471: - z[8,13] <= 0 R472: x[8] + y[13] - z[8,13] <= 1 R473: - y[14] + z[8,14] <= 0 R474: - x[8] + z[8,14] <= 0 R475: - z[8,14] <= 0 R476: x[8] + y[14] - z[8,14] <= 1 R477: - y[15] + z[8,15] <= 0 R478: - x[8] + z[8,15] <= 0 R479: - z[8,15] <= 0 R480: x[8] + y[15] - z[8,15] <= 1 R481: - y[1] + z[9,1] <= 0 R482: - x[9] + z[9,1] <= 0 R483: - z[9,1] <= 0 R484: x[9] + y[1] - z[9,1] <= 1 R485: - y[2] + z[9,2] <= 0 R486: - x[9] + z[9,2] <= 0 R487: - z[9,2] <= 0 R488: x[9] + y[2] - z[9,2] <= 1 R489: - y[3] + z[9,3] <= 0 R490: - x[9] + z[9,3] <= 0 R491: - z[9,3] <= 0 R492: x[9] + y[3] - z[9,3] <= 1 R493: - y[4] + z[9,4] <= 0 R494: - x[9] + z[9,4] <= 0 R495: - z[9,4] <= 0 R496: x[9] + y[4] - z[9,4] <= 1 R497: - y[5] + z[9,5] <= 0 R498: - x[9] + z[9,5] <= 0 R499: - z[9,5] <= 0 R500: x[9] + y[5] - z[9,5] <= 1 R501: - y[6] + z[9,6] <= 0 R502: - x[9] + z[9,6] <= 0 R503: - z[9,6] <= 0 R504: x[9] + y[6] - z[9,6] <= 1 R505: - y[7] + z[9,7] <= 0 R506: - x[9] + z[9,7] <= 0 R507: - z[9,7] <= 0 R508: x[9] + y[7] - z[9,7] <= 1 R509: - y[8] + z[9,8] <= 0 R510: - x[9] + z[9,8] <= 0 R511: - z[9,8] <= 0 R512: x[9] + y[8] - z[9,8] <= 1 R513: - y[9] + z[9,9] <= 0 R514: - x[9] + z[9,9] <= 0 R515: - z[9,9] <= 0 R516: x[9] + y[9] - z[9,9] <= 1 R517: - y[10] + z[9,10] <= 0 R518: - x[9] + z[9,10] <= 0 R519: - z[9,10] <= 0 R520: x[9] + y[10] - z[9,10] <= 1 R521: - y[11] + z[9,11] <= 0 R522: - x[9] + z[9,11] <= 0 R523: - z[9,11] <= 0 R524: x[9] + y[11] - z[9,11] <= 1 R525: - y[12] + z[9,12] <= 0 R526: - x[9] + z[9,12] <= 0 R527: - z[9,12] <= 0 R528: x[9] + y[12] - z[9,12] <= 1 R529: - y[13] + z[9,13] <= 0 R530: - x[9] + z[9,13] <= 0 R531: - z[9,13] <= 0 R532: x[9] + y[13] - z[9,13] <= 1 R533: - y[14] + z[9,14] <= 0 R534: - x[9] + z[9,14] <= 0 R535: - z[9,14] <= 0 R536: x[9] + y[14] - z[9,14] <= 1 R537: - y[15] + z[9,15] <= 0 R538: - x[9] + z[9,15] <= 0 R539: - z[9,15] <= 0 R540: x[9] + y[15] - z[9,15] <= 1 R541: - y[1] + z[10,1] <= 0 R542: - x[10] + z[10,1] <= 0 R543: - z[10,1] <= 0 R544: x[10] + y[1] - z[10,1] <= 1 R545: - y[2] + z[10,2] <= 0 R546: - x[10] + z[10,2] <= 0 R547: - z[10,2] <= 0 R548: x[10] + y[2] - z[10,2] <= 1 R549: - y[3] + z[10,3] <= 0 R550: - x[10] + z[10,3] <= 0 R551: - z[10,3] <= 0 R552: x[10] + y[3] - z[10,3] <= 1 R553: - y[4] + z[10,4] <= 0 R554: - x[10] + z[10,4] <= 0 R555: - z[10,4] <= 0 R556: x[10] + y[4] - z[10,4] <= 1 R557: - y[5] + z[10,5] <= 0 R558: - x[10] + z[10,5] <= 0 R559: - z[10,5] <= 0 R560: x[10] + y[5] - z[10,5] <= 1 R561: - y[6] + z[10,6] <= 0 R562: - x[10] + z[10,6] <= 0 R563: - z[10,6] <= 0 R564: x[10] + y[6] - z[10,6] <= 1 R565: - y[7] + z[10,7] <= 0 R566: - x[10] + z[10,7] <= 0 R567: - z[10,7] <= 0 R568: x[10] + y[7] - z[10,7] <= 1 R569: - y[8] + z[10,8] <= 0 R570: - x[10] + z[10,8] <= 0 R571: - z[10,8] <= 0 R572: x[10] + y[8] - z[10,8] <= 1 R573: - y[9] + z[10,9] <= 0 R574: - x[10] + z[10,9] <= 0 R575: - z[10,9] <= 0 R576: x[10] + y[9] - z[10,9] <= 1 R577: - y[10] + z[10,10] <= 0 R578: - x[10] + z[10,10] <= 0 R579: - z[10,10] <= 0 R580: x[10] + y[10] - z[10,10] <= 1 R581: - y[11] + z[10,11] <= 0 R582: - x[10] + z[10,11] <= 0 R583: - z[10,11] <= 0 R584: x[10] + y[11] - z[10,11] <= 1 R585: - y[12] + z[10,12] <= 0 R586: - x[10] + z[10,12] <= 0 R587: - z[10,12] <= 0 R588: x[10] + y[12] - z[10,12] <= 1 R589: - y[13] + z[10,13] <= 0 R590: - x[10] + z[10,13] <= 0 R591: - z[10,13] <= 0 R592: x[10] + y[13] - z[10,13] <= 1 R593: - y[14] + z[10,14] <= 0 R594: - x[10] + z[10,14] <= 0 R595: - z[10,14] <= 0 R596: x[10] + y[14] - z[10,14] <= 1 R597: - y[15] + z[10,15] <= 0 R598: - x[10] + z[10,15] <= 0 R599: - z[10,15] <= 0 R600: x[10] + y[15] - z[10,15] <= 1 R601: - y[1] + z[11,1] <= 0 R602: - x[11] + z[11,1] <= 0 R603: - z[11,1] <= 0 R604: x[11] + y[1] - z[11,1] <= 1 R605: - y[2] + z[11,2] <= 0 R606: - x[11] + z[11,2] <= 0 R607: - z[11,2] <= 0 R608: x[11] + y[2] - z[11,2] <= 1 R609: - y[3] + z[11,3] <= 0 R610: - x[11] + z[11,3] <= 0 R611: - z[11,3] <= 0 R612: x[11] + y[3] - z[11,3] <= 1 R613: - y[4] + z[11,4] <= 0 R614: - x[11] + z[11,4] <= 0 R615: - z[11,4] <= 0 R616: x[11] + y[4] - z[11,4] <= 1 R617: - y[5] + z[11,5] <= 0 R618: - x[11] + z[11,5] <= 0 R619: - z[11,5] <= 0 R620: x[11] + y[5] - z[11,5] <= 1 R621: - y[6] + z[11,6] <= 0 R622: - x[11] + z[11,6] <= 0 R623: - z[11,6] <= 0 R624: x[11] + y[6] - z[11,6] <= 1 R625: - y[7] + z[11,7] <= 0 R626: - x[11] + z[11,7] <= 0 R627: - z[11,7] <= 0 R628: x[11] + y[7] - z[11,7] <= 1 R629: - y[8] + z[11,8] <= 0 R630: - x[11] + z[11,8] <= 0 R631: - z[11,8] <= 0 R632: x[11] + y[8] - z[11,8] <= 1 R633: - y[9] + z[11,9] <= 0 R634: - x[11] + z[11,9] <= 0 R635: - z[11,9] <= 0 R636: x[11] + y[9] - z[11,9] <= 1 R637: - y[10] + z[11,10] <= 0 R638: - x[11] + z[11,10] <= 0 R639: - z[11,10] <= 0 R640: x[11] + y[10] - z[11,10] <= 1 R641: - y[11] + z[11,11] <= 0 R642: - x[11] + z[11,11] <= 0 R643: - z[11,11] <= 0 R644: x[11] + y[11] - z[11,11] <= 1 R645: - y[12] + z[11,12] <= 0 R646: - x[11] + z[11,12] <= 0 R647: - z[11,12] <= 0 R648: x[11] + y[12] - z[11,12] <= 1 R649: - y[13] + z[11,13] <= 0 R650: - x[11] + z[11,13] <= 0 R651: - z[11,13] <= 0 R652: x[11] + y[13] - z[11,13] <= 1 R653: - y[14] + z[11,14] <= 0 R654: - x[11] + z[11,14] <= 0 R655: - z[11,14] <= 0 R656: x[11] + y[14] - z[11,14] <= 1 R657: - y[15] + z[11,15] <= 0 R658: - x[11] + z[11,15] <= 0 R659: - z[11,15] <= 0 R660: x[11] + y[15] - z[11,15] <= 1 R661: - y[1] + z[12,1] <= 0 R662: - x[12] + z[12,1] <= 0 R663: - z[12,1] <= 0 R664: x[12] + y[1] - z[12,1] <= 1 R665: - y[2] + z[12,2] <= 0 R666: - x[12] + z[12,2] <= 0 R667: - z[12,2] <= 0 R668: x[12] + y[2] - z[12,2] <= 1 R669: - y[3] + z[12,3] <= 0 R670: - x[12] + z[12,3] <= 0 R671: - z[12,3] <= 0 R672: x[12] + y[3] - z[12,3] <= 1 R673: - y[4] + z[12,4] <= 0 R674: - x[12] + z[12,4] <= 0 R675: - z[12,4] <= 0 R676: x[12] + y[4] - z[12,4] <= 1 R677: - y[5] + z[12,5] <= 0 R678: - x[12] + z[12,5] <= 0 R679: - z[12,5] <= 0 R680: x[12] + y[5] - z[12,5] <= 1 R681: - y[6] + z[12,6] <= 0 R682: - x[12] + z[12,6] <= 0 R683: - z[12,6] <= 0 R684: x[12] + y[6] - z[12,6] <= 1 R685: - y[7] + z[12,7] <= 0 R686: - x[12] + z[12,7] <= 0 R687: - z[12,7] <= 0 R688: x[12] + y[7] - z[12,7] <= 1 R689: - y[8] + z[12,8] <= 0 R690: - x[12] + z[12,8] <= 0 R691: - z[12,8] <= 0 R692: x[12] + y[8] - z[12,8] <= 1 R693: - y[9] + z[12,9] <= 0 R694: - x[12] + z[12,9] <= 0 R695: - z[12,9] <= 0 R696: x[12] + y[9] - z[12,9] <= 1 R697: - y[10] + z[12,10] <= 0 R698: - x[12] + z[12,10] <= 0 R699: - z[12,10] <= 0 R700: x[12] + y[10] - z[12,10] <= 1 R701: - y[11] + z[12,11] <= 0 R702: - x[12] + z[12,11] <= 0 R703: - z[12,11] <= 0 R704: x[12] + y[11] - z[12,11] <= 1 R705: - y[12] + z[12,12] <= 0 R706: - x[12] + z[12,12] <= 0 R707: - z[12,12] <= 0 R708: x[12] + y[12] - z[12,12] <= 1 R709: - y[13] + z[12,13] <= 0 R710: - x[12] + z[12,13] <= 0 R711: - z[12,13] <= 0 R712: x[12] + y[13] - z[12,13] <= 1 R713: - y[14] + z[12,14] <= 0 R714: - x[12] + z[12,14] <= 0 R715: - z[12,14] <= 0 R716: x[12] + y[14] - z[12,14] <= 1 R717: - y[15] + z[12,15] <= 0 R718: - x[12] + z[12,15] <= 0 R719: - z[12,15] <= 0 R720: x[12] + y[15] - z[12,15] <= 1 R721: - y[1] + z[13,1] <= 0 R722: - x[13] + z[13,1] <= 0 R723: - z[13,1] <= 0 R724: x[13] + y[1] - z[13,1] <= 1 R725: - y[2] + z[13,2] <= 0 R726: - x[13] + z[13,2] <= 0 R727: - z[13,2] <= 0 R728: x[13] + y[2] - z[13,2] <= 1 R729: - y[3] + z[13,3] <= 0 R730: - x[13] + z[13,3] <= 0 R731: - z[13,3] <= 0 R732: x[13] + y[3] - z[13,3] <= 1 R733: - y[4] + z[13,4] <= 0 R734: - x[13] + z[13,4] <= 0 R735: - z[13,4] <= 0 R736: x[13] + y[4] - z[13,4] <= 1 R737: - y[5] + z[13,5] <= 0 R738: - x[13] + z[13,5] <= 0 R739: - z[13,5] <= 0 R740: x[13] + y[5] - z[13,5] <= 1 R741: - y[6] + z[13,6] <= 0 R742: - x[13] + z[13,6] <= 0 R743: - z[13,6] <= 0 R744: x[13] + y[6] - z[13,6] <= 1 R745: - y[7] + z[13,7] <= 0 R746: - x[13] + z[13,7] <= 0 R747: - z[13,7] <= 0 R748: x[13] + y[7] - z[13,7] <= 1 R749: - y[8] + z[13,8] <= 0 R750: - x[13] + z[13,8] <= 0 R751: - z[13,8] <= 0 R752: x[13] + y[8] - z[13,8] <= 1 R753: - y[9] + z[13,9] <= 0 R754: - x[13] + z[13,9] <= 0 R755: - z[13,9] <= 0 R756: x[13] + y[9] - z[13,9] <= 1 R757: - y[10] + z[13,10] <= 0 R758: - x[13] + z[13,10] <= 0 R759: - z[13,10] <= 0 R760: x[13] + y[10] - z[13,10] <= 1 R761: - y[11] + z[13,11] <= 0 R762: - x[13] + z[13,11] <= 0 R763: - z[13,11] <= 0 R764: x[13] + y[11] - z[13,11] <= 1 R765: - y[12] + z[13,12] <= 0 R766: - x[13] + z[13,12] <= 0 R767: - z[13,12] <= 0 R768: x[13] + y[12] - z[13,12] <= 1 R769: - y[13] + z[13,13] <= 0 R770: - x[13] + z[13,13] <= 0 R771: - z[13,13] <= 0 R772: x[13] + y[13] - z[13,13] <= 1 R773: - y[14] + z[13,14] <= 0 R774: - x[13] + z[13,14] <= 0 R775: - z[13,14] <= 0 R776: x[13] + y[14] - z[13,14] <= 1 R777: - y[15] + z[13,15] <= 0 R778: - x[13] + z[13,15] <= 0 R779: - z[13,15] <= 0 R780: x[13] + y[15] - z[13,15] <= 1 R781: - y[1] + z[14,1] <= 0 R782: - x[14] + z[14,1] <= 0 R783: - z[14,1] <= 0 R784: x[14] + y[1] - z[14,1] <= 1 R785: - y[2] + z[14,2] <= 0 R786: - x[14] + z[14,2] <= 0 R787: - z[14,2] <= 0 R788: x[14] + y[2] - z[14,2] <= 1 R789: - y[3] + z[14,3] <= 0 R790: - x[14] + z[14,3] <= 0 R791: - z[14,3] <= 0 R792: x[14] + y[3] - z[14,3] <= 1 R793: - y[4] + z[14,4] <= 0 R794: - x[14] + z[14,4] <= 0 R795: - z[14,4] <= 0 R796: x[14] + y[4] - z[14,4] <= 1 R797: - y[5] + z[14,5] <= 0 R798: - x[14] + z[14,5] <= 0 R799: - z[14,5] <= 0 R800: x[14] + y[5] - z[14,5] <= 1 R801: - y[6] + z[14,6] <= 0 R802: - x[14] + z[14,6] <= 0 R803: - z[14,6] <= 0 R804: x[14] + y[6] - z[14,6] <= 1 R805: - y[7] + z[14,7] <= 0 R806: - x[14] + z[14,7] <= 0 R807: - z[14,7] <= 0 R808: x[14] + y[7] - z[14,7] <= 1 R809: - y[8] + z[14,8] <= 0 R810: - x[14] + z[14,8] <= 0 R811: - z[14,8] <= 0 R812: x[14] + y[8] - z[14,8] <= 1 R813: - y[9] + z[14,9] <= 0 R814: - x[14] + z[14,9] <= 0 R815: - z[14,9] <= 0 R816: x[14] + y[9] - z[14,9] <= 1 R817: - y[10] + z[14,10] <= 0 R818: - x[14] + z[14,10] <= 0 R819: - z[14,10] <= 0 R820: x[14] + y[10] - z[14,10] <= 1 R821: - y[11] + z[14,11] <= 0 R822: - x[14] + z[14,11] <= 0 R823: - z[14,11] <= 0 R824: x[14] + y[11] - z[14,11] <= 1 R825: - y[12] + z[14,12] <= 0 R826: - x[14] + z[14,12] <= 0 R827: - z[14,12] <= 0 R828: x[14] + y[12] - z[14,12] <= 1 R829: - y[13] + z[14,13] <= 0 R830: - x[14] + z[14,13] <= 0 R831: - z[14,13] <= 0 R832: x[14] + y[13] - z[14,13] <= 1 R833: - y[14] + z[14,14] <= 0 R834: - x[14] + z[14,14] <= 0 R835: - z[14,14] <= 0 R836: x[14] + y[14] - z[14,14] <= 1 R837: - y[15] + z[14,15] <= 0 R838: - x[14] + z[14,15] <= 0 R839: - z[14,15] <= 0 R840: x[14] + y[15] - z[14,15] <= 1 R841: - y[1] + z[15,1] <= 0 R842: - x[15] + z[15,1] <= 0 R843: - z[15,1] <= 0 R844: x[15] + y[1] - z[15,1] <= 1 R845: - y[2] + z[15,2] <= 0 R846: - x[15] + z[15,2] <= 0 R847: - z[15,2] <= 0 R848: x[15] + y[2] - z[15,2] <= 1 R849: - y[3] + z[15,3] <= 0 R850: - x[15] + z[15,3] <= 0 R851: - z[15,3] <= 0 R852: x[15] + y[3] - z[15,3] <= 1 R853: - y[4] + z[15,4] <= 0 R854: - x[15] + z[15,4] <= 0 R855: - z[15,4] <= 0 R856: x[15] + y[4] - z[15,4] <= 1 R857: - y[5] + z[15,5] <= 0 R858: - x[15] + z[15,5] <= 0 R859: - z[15,5] <= 0 R860: x[15] + y[5] - z[15,5] <= 1 R861: - y[6] + z[15,6] <= 0 R862: - x[15] + z[15,6] <= 0 R863: - z[15,6] <= 0 R864: x[15] + y[6] - z[15,6] <= 1 R865: - y[7] + z[15,7] <= 0 R866: - x[15] + z[15,7] <= 0 R867: - z[15,7] <= 0 R868: x[15] + y[7] - z[15,7] <= 1 R869: - y[8] + z[15,8] <= 0 R870: - x[15] + z[15,8] <= 0 R871: - z[15,8] <= 0 R872: x[15] + y[8] - z[15,8] <= 1 R873: - y[9] + z[15,9] <= 0 R874: - x[15] + z[15,9] <= 0 R875: - z[15,9] <= 0 R876: x[15] + y[9] - z[15,9] <= 1 R877: - y[10] + z[15,10] <= 0 R878: - x[15] + z[15,10] <= 0 R879: - z[15,10] <= 0 R880: x[15] + y[10] - z[15,10] <= 1 R881: - y[11] + z[15,11] <= 0 R882: - x[15] + z[15,11] <= 0 R883: - z[15,11] <= 0 R884: x[15] + y[11] - z[15,11] <= 1 R885: - y[12] + z[15,12] <= 0 R886: - x[15] + z[15,12] <= 0 R887: - z[15,12] <= 0 R888: x[15] + y[12] - z[15,12] <= 1 R889: - y[13] + z[15,13] <= 0 R890: - x[15] + z[15,13] <= 0 R891: - z[15,13] <= 0 R892: x[15] + y[13] - z[15,13] <= 1 R893: - y[14] + z[15,14] <= 0 R894: - x[15] + z[15,14] <= 0 R895: - z[15,14] <= 0 R896: x[15] + y[14] - z[15,14] <= 1 R897: - y[15] + z[15,15] <= 0 R898: - x[15] + z[15,15] <= 0 R899: - z[15,15] <= 0 R900: x[15] + y[15] - z[15,15] <= 1 qc0: - z[1,1] + [ x[1] * y[1] ] = 0 qc1: - z[1,2] + [ x[1] * y[2] ] = 0 qc2: - z[1,3] + [ x[1] * y[3] ] = 0 qc3: - z[1,4] + [ x[1] * y[4] ] = 0 qc4: - z[1,5] + [ x[1] * y[5] ] = 0 qc5: - z[1,6] + [ x[1] * y[6] ] = 0 qc6: - z[1,7] + [ x[1] * y[7] ] = 0 qc7: - z[1,8] + [ x[1] * y[8] ] = 0 qc8: - z[1,9] + [ x[1] * y[9] ] = 0 qc9: - z[1,10] + [ x[1] * y[10] ] = 0 qc10: - z[1,11] + [ x[1] * y[11] ] = 0 qc11: - z[1,12] + [ x[1] * y[12] ] = 0 qc12: - z[1,13] + [ x[1] * y[13] ] = 0 qc13: - z[1,14] + [ x[1] * y[14] ] = 0 qc14: - z[1,15] + [ x[1] * y[15] ] = 0 qc15: - z[2,1] + [ x[2] * y[1] ] = 0 qc16: - z[2,2] + [ x[2] * y[2] ] = 0 qc17: - z[2,3] + [ x[2] * y[3] ] = 0 qc18: - z[2,4] + [ x[2] * y[4] ] = 0 qc19: - z[2,5] + [ x[2] * y[5] ] = 0 qc20: - z[2,6] + [ x[2] * y[6] ] = 0 qc21: - z[2,7] + [ x[2] * y[7] ] = 0 qc22: - z[2,8] + [ x[2] * y[8] ] = 0 qc23: - z[2,9] + [ x[2] * y[9] ] = 0 qc24: - z[2,10] + [ x[2] * y[10] ] = 0 qc25: - z[2,11] + [ x[2] * y[11] ] = 0 qc26: - z[2,12] + [ x[2] * y[12] ] = 0 qc27: - z[2,13] + [ x[2] * y[13] ] = 0 qc28: - z[2,14] + [ x[2] * y[14] ] = 0 qc29: - z[2,15] + [ x[2] * y[15] ] = 0 qc30: - z[3,1] + [ x[3] * y[1] ] = 0 qc31: - z[3,2] + [ x[3] * y[2] ] = 0 qc32: - z[3,3] + [ x[3] * y[3] ] = 0 qc33: - z[3,4] + [ x[3] * y[4] ] = 0 qc34: - z[3,5] + [ x[3] * y[5] ] = 0 qc35: - z[3,6] + [ x[3] * y[6] ] = 0 qc36: - z[3,7] + [ x[3] * y[7] ] = 0 qc37: - z[3,8] + [ x[3] * y[8] ] = 0 qc38: - z[3,9] + [ x[3] * y[9] ] = 0 qc39: - z[3,10] + [ x[3] * y[10] ] = 0 qc40: - z[3,11] + [ x[3] * y[11] ] = 0 qc41: - z[3,12] + [ x[3] * y[12] ] = 0 qc42: - z[3,13] + [ x[3] * y[13] ] = 0 qc43: - z[3,14] + [ x[3] * y[14] ] = 0 qc44: - z[3,15] + [ x[3] * y[15] ] = 0 qc45: - z[4,1] + [ x[4] * y[1] ] = 0 qc46: - z[4,2] + [ x[4] * y[2] ] = 0 qc47: - z[4,3] + [ x[4] * y[3] ] = 0 qc48: - z[4,4] + [ x[4] * y[4] ] = 0 qc49: - z[4,5] + [ x[4] * y[5] ] = 0 qc50: - z[4,6] + [ x[4] * y[6] ] = 0 qc51: - z[4,7] + [ x[4] * y[7] ] = 0 qc52: - z[4,8] + [ x[4] * y[8] ] = 0 qc53: - z[4,9] + [ x[4] * y[9] ] = 0 qc54: - z[4,10] + [ x[4] * y[10] ] = 0 qc55: - z[4,11] + [ x[4] * y[11] ] = 0 qc56: - z[4,12] + [ x[4] * y[12] ] = 0 qc57: - z[4,13] + [ x[4] * y[13] ] = 0 qc58: - z[4,14] + [ x[4] * y[14] ] = 0 qc59: - z[4,15] + [ x[4] * y[15] ] = 0 qc60: - z[5,1] + [ x[5] * y[1] ] = 0 qc61: - z[5,2] + [ x[5] * y[2] ] = 0 qc62: - z[5,3] + [ x[5] * y[3] ] = 0 qc63: - z[5,4] + [ x[5] * y[4] ] = 0 qc64: - z[5,5] + [ x[5] * y[5] ] = 0 qc65: - z[5,6] + [ x[5] * y[6] ] = 0 qc66: - z[5,7] + [ x[5] * y[7] ] = 0 qc67: - z[5,8] + [ x[5] * y[8] ] = 0 qc68: - z[5,9] + [ x[5] * y[9] ] = 0 qc69: - z[5,10] + [ x[5] * y[10] ] = 0 qc70: - z[5,11] + [ x[5] * y[11] ] = 0 qc71: - z[5,12] + [ x[5] * y[12] ] = 0 qc72: - z[5,13] + [ x[5] * y[13] ] = 0 qc73: - z[5,14] + [ x[5] * y[14] ] = 0 qc74: - z[5,15] + [ x[5] * y[15] ] = 0 qc75: - z[6,1] + [ x[6] * y[1] ] = 0 qc76: - z[6,2] + [ x[6] * y[2] ] = 0 qc77: - z[6,3] + [ x[6] * y[3] ] = 0 qc78: - z[6,4] + [ x[6] * y[4] ] = 0 qc79: - z[6,5] + [ x[6] * y[5] ] = 0 qc80: - z[6,6] + [ x[6] * y[6] ] = 0 qc81: - z[6,7] + [ x[6] * y[7] ] = 0 qc82: - z[6,8] + [ x[6] * y[8] ] = 0 qc83: - z[6,9] + [ x[6] * y[9] ] = 0 qc84: - z[6,10] + [ x[6] * y[10] ] = 0 qc85: - z[6,11] + [ x[6] * y[11] ] = 0 qc86: - z[6,12] + [ x[6] * y[12] ] = 0 qc87: - z[6,13] + [ x[6] * y[13] ] = 0 qc88: - z[6,14] + [ x[6] * y[14] ] = 0 qc89: - z[6,15] + [ x[6] * y[15] ] = 0 qc90: - z[7,1] + [ x[7] * y[1] ] = 0 qc91: - z[7,2] + [ x[7] * y[2] ] = 0 qc92: - z[7,3] + [ x[7] * y[3] ] = 0 qc93: - z[7,4] + [ x[7] * y[4] ] = 0 qc94: - z[7,5] + [ x[7] * y[5] ] = 0 qc95: - z[7,6] + [ x[7] * y[6] ] = 0 qc96: - z[7,7] + [ x[7] * y[7] ] = 0 qc97: - z[7,8] + [ x[7] * y[8] ] = 0 qc98: - z[7,9] + [ x[7] * y[9] ] = 0 qc99: - z[7,10] + [ x[7] * y[10] ] = 0 qc100: - z[7,11] + [ x[7] * y[11] ] = 0 qc101: - z[7,12] + [ x[7] * y[12] ] = 0 qc102: - z[7,13] + [ x[7] * y[13] ] = 0 qc103: - z[7,14] + [ x[7] * y[14] ] = 0 qc104: - z[7,15] + [ x[7] * y[15] ] = 0 qc105: - z[8,1] + [ x[8] * y[1] ] = 0 qc106: - z[8,2] + [ x[8] * y[2] ] = 0 qc107: - z[8,3] + [ x[8] * y[3] ] = 0 qc108: - z[8,4] + [ x[8] * y[4] ] = 0 qc109: - z[8,5] + [ x[8] * y[5] ] = 0 qc110: - z[8,6] + [ x[8] * y[6] ] = 0 qc111: - z[8,7] + [ x[8] * y[7] ] = 0 qc112: - z[8,8] + [ x[8] * y[8] ] = 0 qc113: - z[8,9] + [ x[8] * y[9] ] = 0 qc114: - z[8,10] + [ x[8] * y[10] ] = 0 qc115: - z[8,11] + [ x[8] * y[11] ] = 0 qc116: - z[8,12] + [ x[8] * y[12] ] = 0 qc117: - z[8,13] + [ x[8] * y[13] ] = 0 qc118: - z[8,14] + [ x[8] * y[14] ] = 0 qc119: - z[8,15] + [ x[8] * y[15] ] = 0 qc120: - z[9,1] + [ x[9] * y[1] ] = 0 qc121: - z[9,2] + [ x[9] * y[2] ] = 0 qc122: - z[9,3] + [ x[9] * y[3] ] = 0 qc123: - z[9,4] + [ x[9] * y[4] ] = 0 qc124: - z[9,5] + [ x[9] * y[5] ] = 0 qc125: - z[9,6] + [ x[9] * y[6] ] = 0 qc126: - z[9,7] + [ x[9] * y[7] ] = 0 qc127: - z[9,8] + [ x[9] * y[8] ] = 0 qc128: - z[9,9] + [ x[9] * y[9] ] = 0 qc129: - z[9,10] + [ x[9] * y[10] ] = 0 qc130: - z[9,11] + [ x[9] * y[11] ] = 0 qc131: - z[9,12] + [ x[9] * y[12] ] = 0 qc132: - z[9,13] + [ x[9] * y[13] ] = 0 qc133: - z[9,14] + [ x[9] * y[14] ] = 0 qc134: - z[9,15] + [ x[9] * y[15] ] = 0 qc135: - z[10,1] + [ x[10] * y[1] ] = 0 qc136: - z[10,2] + [ x[10] * y[2] ] = 0 qc137: - z[10,3] + [ x[10] * y[3] ] = 0 qc138: - z[10,4] + [ x[10] * y[4] ] = 0 qc139: - z[10,5] + [ x[10] * y[5] ] = 0 qc140: - z[10,6] + [ x[10] * y[6] ] = 0 qc141: - z[10,7] + [ x[10] * y[7] ] = 0 qc142: - z[10,8] + [ x[10] * y[8] ] = 0 qc143: - z[10,9] + [ x[10] * y[9] ] = 0 qc144: - z[10,10] + [ x[10] * y[10] ] = 0 qc145: - z[10,11] + [ x[10] * y[11] ] = 0 qc146: - z[10,12] + [ x[10] * y[12] ] = 0 qc147: - z[10,13] + [ x[10] * y[13] ] = 0 qc148: - z[10,14] + [ x[10] * y[14] ] = 0 qc149: - z[10,15] + [ x[10] * y[15] ] = 0 qc150: - z[11,1] + [ x[11] * y[1] ] = 0 qc151: - z[11,2] + [ x[11] * y[2] ] = 0 qc152: - z[11,3] + [ x[11] * y[3] ] = 0 qc153: - z[11,4] + [ x[11] * y[4] ] = 0 qc154: - z[11,5] + [ x[11] * y[5] ] = 0 qc155: - z[11,6] + [ x[11] * y[6] ] = 0 qc156: - z[11,7] + [ x[11] * y[7] ] = 0 qc157: - z[11,8] + [ x[11] * y[8] ] = 0 qc158: - z[11,9] + [ x[11] * y[9] ] = 0 qc159: - z[11,10] + [ x[11] * y[10] ] = 0 qc160: - z[11,11] + [ x[11] * y[11] ] = 0 qc161: - z[11,12] + [ x[11] * y[12] ] = 0 qc162: - z[11,13] + [ x[11] * y[13] ] = 0 qc163: - z[11,14] + [ x[11] * y[14] ] = 0 qc164: - z[11,15] + [ x[11] * y[15] ] = 0 qc165: - z[12,1] + [ x[12] * y[1] ] = 0 qc166: - z[12,2] + [ x[12] * y[2] ] = 0 qc167: - z[12,3] + [ x[12] * y[3] ] = 0 qc168: - z[12,4] + [ x[12] * y[4] ] = 0 qc169: - z[12,5] + [ x[12] * y[5] ] = 0 qc170: - z[12,6] + [ x[12] * y[6] ] = 0 qc171: - z[12,7] + [ x[12] * y[7] ] = 0 qc172: - z[12,8] + [ x[12] * y[8] ] = 0 qc173: - z[12,9] + [ x[12] * y[9] ] = 0 qc174: - z[12,10] + [ x[12] * y[10] ] = 0 qc175: - z[12,11] + [ x[12] * y[11] ] = 0 qc176: - z[12,12] + [ x[12] * y[12] ] = 0 qc177: - z[12,13] + [ x[12] * y[13] ] = 0 qc178: - z[12,14] + [ x[12] * y[14] ] = 0 qc179: - z[12,15] + [ x[12] * y[15] ] = 0 qc180: - z[13,1] + [ x[13] * y[1] ] = 0 qc181: - z[13,2] + [ x[13] * y[2] ] = 0 qc182: - z[13,3] + [ x[13] * y[3] ] = 0 qc183: - z[13,4] + [ x[13] * y[4] ] = 0 qc184: - z[13,5] + [ x[13] * y[5] ] = 0 qc185: - z[13,6] + [ x[13] * y[6] ] = 0 qc186: - z[13,7] + [ x[13] * y[7] ] = 0 qc187: - z[13,8] + [ x[13] * y[8] ] = 0 qc188: - z[13,9] + [ x[13] * y[9] ] = 0 qc189: - z[13,10] + [ x[13] * y[10] ] = 0 qc190: - z[13,11] + [ x[13] * y[11] ] = 0 qc191: - z[13,12] + [ x[13] * y[12] ] = 0 qc192: - z[13,13] + [ x[13] * y[13] ] = 0 qc193: - z[13,14] + [ x[13] * y[14] ] = 0 qc194: - z[13,15] + [ x[13] * y[15] ] = 0 qc195: - z[14,1] + [ x[14] * y[1] ] = 0 qc196: - z[14,2] + [ x[14] * y[2] ] = 0 qc197: - z[14,3] + [ x[14] * y[3] ] = 0 qc198: - z[14,4] + [ x[14] * y[4] ] = 0 qc199: - z[14,5] + [ x[14] * y[5] ] = 0 qc200: - z[14,6] + [ x[14] * y[6] ] = 0 qc201: - z[14,7] + [ x[14] * y[7] ] = 0 qc202: - z[14,8] + [ x[14] * y[8] ] = 0 qc203: - z[14,9] + [ x[14] * y[9] ] = 0 qc204: - z[14,10] + [ x[14] * y[10] ] = 0 qc205: - z[14,11] + [ x[14] * y[11] ] = 0 qc206: - z[14,12] + [ x[14] * y[12] ] = 0 qc207: - z[14,13] + [ x[14] * y[13] ] = 0 qc208: - z[14,14] + [ x[14] * y[14] ] = 0 qc209: - z[14,15] + [ x[14] * y[15] ] = 0 qc210: - z[15,1] + [ x[15] * y[1] ] = 0 qc211: - z[15,2] + [ x[15] * y[2] ] = 0 qc212: - z[15,3] + [ x[15] * y[3] ] = 0 qc213: - z[15,4] + [ x[15] * y[4] ] = 0 qc214: - z[15,5] + [ x[15] * y[5] ] = 0 qc215: - z[15,6] + [ x[15] * y[6] ] = 0 qc216: - z[15,7] + [ x[15] * y[7] ] = 0 qc217: - z[15,8] + [ x[15] * y[8] ] = 0 qc218: - z[15,9] + [ x[15] * y[9] ] = 0 qc219: - z[15,10] + [ x[15] * y[10] ] = 0 qc220: - z[15,11] + [ x[15] * y[11] ] = 0 qc221: - z[15,12] + [ x[15] * y[12] ] = 0 qc222: - z[15,13] + [ x[15] * y[13] ] = 0 qc223: - z[15,14] + [ x[15] * y[14] ] = 0 qc224: - z[15,15] + [ x[15] * y[15] ] = 0Bounds x[1] <= 1 x[2] <= 1 x[3] <= 1 x[4] <= 1 x[5] <= 1 x[6] <= 1 x[7] <= 1 x[8] <= 1 x[9] <= 1 x[10] <= 1 x[11] <= 1 x[12] <= 1 x[13] <= 1 x[14] <= 1 x[15] <= 1 y[1] <= 1 y[2] <= 1 y[3] <= 1 y[4] <= 1 y[5] <= 1 y[6] <= 1 y[7] <= 1 y[8] <= 1 y[9] <= 1 y[10] <= 1 y[11] <= 1 y[12] <= 1 y[13] <= 1 y[14] <= 1 y[15] <= 1 z[1,1] <= 1 z[1,2] <= 1 z[1,3] <= 1 z[1,4] <= 1 z[1,5] <= 1 z[1,6] <= 1 z[1,7] <= 1 z[1,8] <= 1 z[1,9] <= 1 z[1,10] <= 1 z[1,11] <= 1 z[1,12] <= 1 z[1,13] <= 1 z[1,14] <= 1 z[1,15] <= 1 z[2,1] <= 1 z[2,2] <= 1 z[2,3] <= 1 z[2,4] <= 1 z[2,5] <= 1 z[2,6] <= 1 z[2,7] <= 1 z[2,8] <= 1 z[2,9] <= 1 z[2,10] <= 1 z[2,11] <= 1 z[2,12] <= 1 z[2,13] <= 1 z[2,14] <= 1 z[2,15] <= 1 z[3,1] <= 1 z[3,2] <= 1 z[3,3] <= 1 z[3,4] <= 1 z[3,5] <= 1 z[3,6] <= 1 z[3,7] <= 1 z[3,8] <= 1 z[3,9] <= 1 z[3,10] <= 1 z[3,11] <= 1 z[3,12] <= 1 z[3,13] <= 1 z[3,14] <= 1 z[3,15] <= 1 z[4,1] <= 1 z[4,2] <= 1 z[4,3] <= 1 z[4,4] <= 1 z[4,5] <= 1 z[4,6] <= 1 z[4,7] <= 1 z[4,8] <= 1 z[4,9] <= 1 z[4,10] <= 1 z[4,11] <= 1 z[4,12] <= 1 z[4,13] <= 1 z[4,14] <= 1 z[4,15] <= 1 z[5,1] <= 1 z[5,2] <= 1 z[5,3] <= 1 z[5,4] <= 1 z[5,5] <= 1 z[5,6] <= 1 z[5,7] <= 1 z[5,8] <= 1 z[5,9] <= 1 z[5,10] <= 1 z[5,11] <= 1 z[5,12] <= 1 z[5,13] <= 1 z[5,14] <= 1 z[5,15] <= 1 z[6,1] <= 1 z[6,2] <= 1 z[6,3] <= 1 z[6,4] <= 1 z[6,5] <= 1 z[6,6] <= 1 z[6,7] <= 1 z[6,8] <= 1 z[6,9] <= 1 z[6,10] <= 1 z[6,11] <= 1 z[6,12] <= 1 z[6,13] <= 1 z[6,14] <= 1 z[6,15] <= 1 z[7,1] <= 1 z[7,2] <= 1 z[7,3] <= 1 z[7,4] <= 1 z[7,5] <= 1 z[7,6] <= 1 z[7,7] <= 1 z[7,8] <= 1 z[7,9] <= 1 z[7,10] <= 1 z[7,11] <= 1 z[7,12] <= 1 z[7,13] <= 1 z[7,14] <= 1 z[7,15] <= 1 z[8,1] <= 1 z[8,2] <= 1 z[8,3] <= 1 z[8,4] <= 1 z[8,5] <= 1 z[8,6] <= 1 z[8,7] <= 1 z[8,8] <= 1 z[8,9] <= 1 z[8,10] <= 1 z[8,11] <= 1 z[8,12] <= 1 z[8,13] <= 1 z[8,14] <= 1 z[8,15] <= 1 z[9,1] <= 1 z[9,2] <= 1 z[9,3] <= 1 z[9,4] <= 1 z[9,5] <= 1 z[9,6] <= 1 z[9,7] <= 1 z[9,8] <= 1 z[9,9] <= 1 z[9,10] <= 1 z[9,11] <= 1 z[9,12] <= 1 z[9,13] <= 1 z[9,14] <= 1 z[9,15] <= 1 z[10,1] <= 1 z[10,2] <= 1 z[10,3] <= 1 z[10,4] <= 1 z[10,5] <= 1 z[10,6] <= 1 z[10,7] <= 1 z[10,8] <= 1 z[10,9] <= 1 z[10,10] <= 1 z[10,11] <= 1 z[10,12] <= 1 z[10,13] <= 1 z[10,14] <= 1 z[10,15] <= 1 z[11,1] <= 1 z[11,2] <= 1 z[11,3] <= 1 z[11,4] <= 1 z[11,5] <= 1 z[11,6] <= 1 z[11,7] <= 1 z[11,8] <= 1 z[11,9] <= 1 z[11,10] <= 1 z[11,11] <= 1 z[11,12] <= 1 z[11,13] <= 1 z[11,14] <= 1 z[11,15] <= 1 z[12,1] <= 1 z[12,2] <= 1 z[12,3] <= 1 z[12,4] <= 1 z[12,5] <= 1 z[12,6] <= 1 z[12,7] <= 1 z[12,8] <= 1 z[12,9] <= 1 z[12,10] <= 1 z[12,11] <= 1 z[12,12] <= 1 z[12,13] <= 1 z[12,14] <= 1 z[12,15] <= 1 z[13,1] <= 1 z[13,2] <= 1 z[13,3] <= 1 z[13,4] <= 1 z[13,5] <= 1 z[13,6] <= 1 z[13,7] <= 1 z[13,8] <= 1 z[13,9] <= 1 z[13,10] <= 1 z[13,11] <= 1 z[13,12] <= 1 z[13,13] <= 1 z[13,14] <= 1 z[13,15] <= 1 z[14,1] <= 1 z[14,2] <= 1 z[14,3] <= 1 z[14,4] <= 1 z[14,5] <= 1 z[14,6] <= 1 z[14,7] <= 1 z[14,8] <= 1 z[14,9] <= 1 z[14,10] <= 1 z[14,11] <= 1 z[14,12] <= 1 z[14,13] <= 1 z[14,14] <= 1 z[14,15] <= 1 z[15,1] <= 1 z[15,2] <= 1 z[15,3] <= 1 z[15,4] <= 1 z[15,5] <= 1 z[15,6] <= 1 z[15,7] <= 1 z[15,8] <= 1 z[15,9] <= 1 z[15,10] <= 1 z[15,11] <= 1 z[15,12] <= 1 z[15,13] <= 1 z[15,14] <= 1 z[15,15] <= 1End

And here is my manual McCormick relaxation:

Maximize  1.29 x[1] + 3.91 x[2] - 1.24 x[3] - 0.68 x[4] - 2.74 x[5] - 2.09 x[6]   + 4.72 x[7] - 1.2 x[8] + 4.61 x[9] + 4.14 x[10] + 0.96 x[11] - 2.4 x[12]   + 4.81 x[13] - 0.04 x[14] - 0.85 x[15] + 1.29 y[1] + 3.91 y[2]   - 1.24 y[3] - 0.68 y[4] - 2.74 y[5] - 2.09 y[6] + 4.72 y[7] - 1.2 y[8]   + 4.61 y[9] + 4.14 y[10] + 0.96 y[11] - 2.4 y[12] + 4.81 y[13]   - 0.04 y[14] - 0.85 y[15] + 4.01 z[1,1] - 3.87 z[1,2] - 0.31 z[1,3]   - 2.53 z[1,4] + 0.44 z[1,5] + 0.74 z[1,6] - 4.87 z[1,7] - 2.83 z[1,8]   - 2.21 z[1,9] + 4.16 z[1,10] + 2.66 z[1,11] - 3.4 z[1,12] + 2.97 z[1,13]   - 3.61 z[1,14] + 1.17 z[1,15] - 3.73 z[2,1] - 4.98 z[2,2] + 3.71 z[2,3]   - 2.91 z[2,4] - 2.85 z[2,5] + 4.82 z[2,6] + 3.72 z[2,7] - 2.11 z[2,8]   + 4.61 z[2,9] + 0.39 z[2,10] + 1.78 z[2,11] - 2.95 z[2,12]   + 4.41 z[2,13] + 1.91 z[2,14] + 4.67 z[2,15] + 3.94 z[3,1] - 2.01 z[3,2]   - 1.39 z[3,3] - 3.34 z[3,4] - 3.54 z[3,5] - 4.35 z[3,6] - 1.99 z[3,7]   + 1.03 z[3,8] - 4.97 z[3,9] + 1.78 z[3,10] - 1.62 z[3,11] - 1.9 z[3,12]   + 3.19 z[3,13] - 0.19 z[3,14] - 1.84 z[3,15] - 0.19 z[4,1] + 2.05 z[4,2]   - 4.43 z[4,3] + 4.75 z[4,4] - 4.77 z[4,5] + 2.5 z[4,6] + 3.45 z[4,7]   - 4.82 z[4,8] + 2.88 z[4,9] - 1.34 z[4,10] + 0.79 z[4,11] - 4.91 z[4,12]   - 4.53 z[4,13] - 3.19 z[4,14] + 4.55 z[4,15] - 3.03 z[5,1] + 2.56 z[5,2]   + 4.3 z[5,3] + 4.42 z[5,4] - 1.56 z[5,5] - 1.45 z[5,6] + 0.25 z[5,7]   + 2.76 z[5,8] - 3.92 z[5,9] + 2.48 z[5,10] + 2.97 z[5,11] + 3.6 z[5,12]   - 4.63 z[5,13] + 4.46 z[5,14] - 4.09 z[5,15] - 1.59 z[6,1] + 1.11 z[6,2]   + 4.18 z[6,3] - 1.6 z[6,4] + 4.24 z[6,5] + 0.45 z[6,6] - 1.88 z[6,7]   - 1.83 z[6,8] - 3.23 z[6,9] - 4.22 z[6,10] - 3.51 z[6,11] + 1.89 z[6,12]   + 4.97 z[6,13] - 3.38 z[6,14] - 4.51 z[6,15] + 4.87 z[7,1] + 0.34 z[7,2]   - 0.94 z[7,3] - 2.63 z[7,4] + 0.94 z[7,5] + 3.26 z[7,6] - 0.44 z[7,7]   - 0.78 z[7,8] - 4.44 z[7,9] + 4.16 z[7,10] - 4.67 z[7,11] - 0.06 z[7,12]   + 3.38 z[7,13] - 3.69 z[7,14] + 2.32 z[7,15] + 4.5 z[8,1] + 1.3 z[8,2]   + 2.88 z[8,3] - 3.93 z[8,4] - 0.65 z[8,5] - 3.51 z[8,6] + 3.45 z[8,7]   - 2.05 z[8,8] - 0.47 z[8,9] + 4.99 z[8,10] + 3.52 z[8,11] + 4.76 z[8,12]   - 0.46 z[8,13] - 0.12 z[8,14] + 2.3 z[8,15] - 0.21 z[9,1] - 2.09 z[9,2]   - 0.96 z[9,3] - 3.53 z[9,4] - 1.23 z[9,5] + 4.88 z[9,6] + 4.6 z[9,7]   + 1.27 z[9,8] - 0.01 z[9,9] - 1.62 z[9,10] - 4.11 z[9,11] - 2.28 z[9,12]   + 2.82 z[9,13] + 3.67 z[9,14] - 1.39 z[9,15] + 2.86 z[10,1]   + 2.75 z[10,2] + 1.95 z[10,3] + 1.64 z[10,4] + 2.6 z[10,5]   - 1.37 z[10,6] + 2.04 z[10,7] - 2.19 z[10,8] - 0.14 z[10,9]   + 2.7 z[10,10] + 1.91 z[10,11] - 2.06 z[10,12] + 4.46 z[10,13]   + 1.5 z[10,14] + 0.81 z[10,15] - 4.88 z[11,1] + 0.47 z[11,2]   - 2.49 z[11,3] + 1.72 z[11,4] - 0.37 z[11,5] + 3.17 z[11,6]   + 1.47 z[11,7] + 2.98 z[11,8] - 1.52 z[11,9] + 1.44 z[11,10]   + 2.38 z[11,11] + 3.28 z[11,12] - 1.5 z[11,13] + 3.43 z[11,14]   + 3.7 z[11,15] + 1.88 z[12,1] + 4.76 z[12,2] + 4.57 z[12,3]   + 0.18 z[12,4] + 0.29 z[12,5] - 3.34 z[12,6] + 3.37 z[12,7]   + 4.37 z[12,8] - 0.23 z[12,9] + 1.91 z[12,10] + 2.2 z[12,11]   + 2.3 z[12,12] - 3.28 z[12,13] + 2.8 z[12,14] + 0.81 z[12,15]   + 1.66 z[13,1] - 0.79 z[13,2] + 1.24 z[13,3] + 2.75 z[13,4]   + 1.37 z[13,5] + 2.2 z[13,6] - 4.72 z[13,7] - 3.4 z[13,8] - 0.59 z[13,9]   + 1.5 z[13,10] - 2.81 z[13,11] + 1.86 z[13,12] + 1.31 z[13,13]   - 4.58 z[13,14] - 0.28 z[13,15] - 2.74 z[14,1] - 4.46 z[14,2]   - 3.66 z[14,3] - 1.83 z[14,4] - 3.18 z[14,5] - 3.07 z[14,6]   - 4.64 z[14,7] - 0.35 z[14,8] - 1.2 z[14,9] + 1.12 z[14,10]   + 0.9 z[14,11] - 2.62 z[14,12] + 4.03 z[14,13] - 4.99 z[14,14]   - 0.95 z[14,15] - 2.21 z[15,1] - 0.9 z[15,2] - 3.85 z[15,3]   + 3.31 z[15,4] - 1.26 z[15,5] - 4.64 z[15,6] + 1.14 z[15,7]   - 4.05 z[15,8] + 0.45 z[15,9] - 1.61 z[15,10] + 0.81 z[15,11]   + 4.58 z[15,12] + 3.19 z[15,13] - 0.81 z[15,14] + 3.13 z[15,15]Subject To R0: x[1] + x[2] + x[3] <= 0.48 R1: x[1] + y[1] - z[1,1] <= 1 R2: x[1] + y[2] - z[1,2] <= 1 R3: x[1] + y[3] - z[1,3] <= 1 R4: x[1] + y[4] - z[1,4] <= 1 R5: x[1] + y[5] - z[1,5] <= 1 R6: x[1] + y[6] - z[1,6] <= 1 R7: x[1] + y[7] - z[1,7] <= 1 R8: x[1] + y[8] - z[1,8] <= 1 R9: x[1] + y[9] - z[1,9] <= 1 R10: x[1] + y[10] - z[1,10] <= 1 R11: x[1] + y[11] - z[1,11] <= 1 R12: x[1] + y[12] - z[1,12] <= 1 R13: x[1] + y[13] - z[1,13] <= 1 R14: x[1] + y[14] - z[1,14] <= 1 R15: x[1] + y[15] - z[1,15] <= 1 R16: x[2] + y[1] - z[2,1] <= 1 R17: x[2] + y[2] - z[2,2] <= 1 R18: x[2] + y[3] - z[2,3] <= 1 R19: x[2] + y[4] - z[2,4] <= 1 R20: x[2] + y[5] - z[2,5] <= 1 R21: x[2] + y[6] - z[2,6] <= 1 R22: x[2] + y[7] - z[2,7] <= 1 R23: x[2] + y[8] - z[2,8] <= 1 R24: x[2] + y[9] - z[2,9] <= 1 R25: x[2] + y[10] - z[2,10] <= 1 R26: x[2] + y[11] - z[2,11] <= 1 R27: x[2] + y[12] - z[2,12] <= 1 R28: x[2] + y[13] - z[2,13] <= 1 R29: x[2] + y[14] - z[2,14] <= 1 R30: x[2] + y[15] - z[2,15] <= 1 R31: x[3] + y[1] - z[3,1] <= 1 R32: x[3] + y[2] - z[3,2] <= 1 R33: x[3] + y[3] - z[3,3] <= 1 R34: x[3] + y[4] - z[3,4] <= 1 R35: x[3] + y[5] - z[3,5] <= 1 R36: x[3] + y[6] - z[3,6] <= 1 R37: x[3] + y[7] - z[3,7] <= 1 R38: x[3] + y[8] - z[3,8] <= 1 R39: x[3] + y[9] - z[3,9] <= 1 R40: x[3] + y[10] - z[3,10] <= 1 R41: x[3] + y[11] - z[3,11] <= 1 R42: x[3] + y[12] - z[3,12] <= 1 R43: x[3] + y[13] - z[3,13] <= 1 R44: x[3] + y[14] - z[3,14] <= 1 R45: x[3] + y[15] - z[3,15] <= 1 R46: x[4] + y[1] - z[4,1] <= 1 R47: x[4] + y[2] - z[4,2] <= 1 R48: x[4] + y[3] - z[4,3] <= 1 R49: x[4] + y[4] - z[4,4] <= 1 R50: x[4] + y[5] - z[4,5] <= 1 R51: x[4] + y[6] - z[4,6] <= 1 R52: x[4] + y[7] - z[4,7] <= 1 R53: x[4] + y[8] - z[4,8] <= 1 R54: x[4] + y[9] - z[4,9] <= 1 R55: x[4] + y[10] - z[4,10] <= 1 R56: x[4] + y[11] - z[4,11] <= 1 R57: x[4] + y[12] - z[4,12] <= 1 R58: x[4] + y[13] - z[4,13] <= 1 R59: x[4] + y[14] - z[4,14] <= 1 R60: x[4] + y[15] - z[4,15] <= 1 R61: x[5] + y[1] - z[5,1] <= 1 R62: x[5] + y[2] - z[5,2] <= 1 R63: x[5] + y[3] - z[5,3] <= 1 R64: x[5] + y[4] - z[5,4] <= 1 R65: x[5] + y[5] - z[5,5] <= 1 R66: x[5] + y[6] - z[5,6] <= 1 R67: x[5] + y[7] - z[5,7] <= 1 R68: x[5] + y[8] - z[5,8] <= 1 R69: x[5] + y[9] - z[5,9] <= 1 R70: x[5] + y[10] - z[5,10] <= 1 R71: x[5] + y[11] - z[5,11] <= 1 R72: x[5] + y[12] - z[5,12] <= 1 R73: x[5] + y[13] - z[5,13] <= 1 R74: x[5] + y[14] - z[5,14] <= 1 R75: x[5] + y[15] - z[5,15] <= 1 R76: x[6] + y[1] - z[6,1] <= 1 R77: x[6] + y[2] - z[6,2] <= 1 R78: x[6] + y[3] - z[6,3] <= 1 R79: x[6] + y[4] - z[6,4] <= 1 R80: x[6] + y[5] - z[6,5] <= 1 R81: x[6] + y[6] - z[6,6] <= 1 R82: x[6] + y[7] - z[6,7] <= 1 R83: x[6] + y[8] - z[6,8] <= 1 R84: x[6] + y[9] - z[6,9] <= 1 R85: x[6] + y[10] - z[6,10] <= 1 R86: x[6] + y[11] - z[6,11] <= 1 R87: x[6] + y[12] - z[6,12] <= 1 R88: x[6] + y[13] - z[6,13] <= 1 R89: x[6] + y[14] - z[6,14] <= 1 R90: x[6] + y[15] - z[6,15] <= 1 R91: x[7] + y[1] - z[7,1] <= 1 R92: x[7] + y[2] - z[7,2] <= 1 R93: x[7] + y[3] - z[7,3] <= 1 R94: x[7] + y[4] - z[7,4] <= 1 R95: x[7] + y[5] - z[7,5] <= 1 R96: x[7] + y[6] - z[7,6] <= 1 R97: x[7] + y[7] - z[7,7] <= 1 R98: x[7] + y[8] - z[7,8] <= 1 R99: x[7] + y[9] - z[7,9] <= 1 R100: x[7] + y[10] - z[7,10] <= 1 R101: x[7] + y[11] - z[7,11] <= 1 R102: x[7] + y[12] - z[7,12] <= 1 R103: x[7] + y[13] - z[7,13] <= 1 R104: x[7] + y[14] - z[7,14] <= 1 R105: x[7] + y[15] - z[7,15] <= 1 R106: x[8] + y[1] - z[8,1] <= 1 R107: x[8] + y[2] - z[8,2] <= 1 R108: x[8] + y[3] - z[8,3] <= 1 R109: x[8] + y[4] - z[8,4] <= 1 R110: x[8] + y[5] - z[8,5] <= 1 R111: x[8] + y[6] - z[8,6] <= 1 R112: x[8] + y[7] - z[8,7] <= 1 R113: x[8] + y[8] - z[8,8] <= 1 R114: x[8] + y[9] - z[8,9] <= 1 R115: x[8] + y[10] - z[8,10] <= 1 R116: x[8] + y[11] - z[8,11] <= 1 R117: x[8] + y[12] - z[8,12] <= 1 R118: x[8] + y[13] - z[8,13] <= 1 R119: x[8] + y[14] - z[8,14] <= 1 R120: x[8] + y[15] - z[8,15] <= 1 R121: x[9] + y[1] - z[9,1] <= 1 R122: x[9] + y[2] - z[9,2] <= 1 R123: x[9] + y[3] - z[9,3] <= 1 R124: x[9] + y[4] - z[9,4] <= 1 R125: x[9] + y[5] - z[9,5] <= 1 R126: x[9] + y[6] - z[9,6] <= 1 R127: x[9] + y[7] - z[9,7] <= 1 R128: x[9] + y[8] - z[9,8] <= 1 R129: x[9] + y[9] - z[9,9] <= 1 R130: x[9] + y[10] - z[9,10] <= 1 R131: x[9] + y[11] - z[9,11] <= 1 R132: x[9] + y[12] - z[9,12] <= 1 R133: x[9] + y[13] - z[9,13] <= 1 R134: x[9] + y[14] - z[9,14] <= 1 R135: x[9] + y[15] - z[9,15] <= 1 R136: x[10] + y[1] - z[10,1] <= 1 R137: x[10] + y[2] - z[10,2] <= 1 R138: x[10] + y[3] - z[10,3] <= 1 R139: x[10] + y[4] - z[10,4] <= 1 R140: x[10] + y[5] - z[10,5] <= 1 R141: x[10] + y[6] - z[10,6] <= 1 R142: x[10] + y[7] - z[10,7] <= 1 R143: x[10] + y[8] - z[10,8] <= 1 R144: x[10] + y[9] - z[10,9] <= 1 R145: x[10] + y[10] - z[10,10] <= 1 R146: x[10] + y[11] - z[10,11] <= 1 R147: x[10] + y[12] - z[10,12] <= 1 R148: x[10] + y[13] - z[10,13] <= 1 R149: x[10] + y[14] - z[10,14] <= 1 R150: x[10] + y[15] - z[10,15] <= 1 R151: x[11] + y[1] - z[11,1] <= 1 R152: x[11] + y[2] - z[11,2] <= 1 R153: x[11] + y[3] - z[11,3] <= 1 R154: x[11] + y[4] - z[11,4] <= 1 R155: x[11] + y[5] - z[11,5] <= 1 R156: x[11] + y[6] - z[11,6] <= 1 R157: x[11] + y[7] - z[11,7] <= 1 R158: x[11] + y[8] - z[11,8] <= 1 R159: x[11] + y[9] - z[11,9] <= 1 R160: x[11] + y[10] - z[11,10] <= 1 R161: x[11] + y[11] - z[11,11] <= 1 R162: x[11] + y[12] - z[11,12] <= 1 R163: x[11] + y[13] - z[11,13] <= 1 R164: x[11] + y[14] - z[11,14] <= 1 R165: x[11] + y[15] - z[11,15] <= 1 R166: x[12] + y[1] - z[12,1] <= 1 R167: x[12] + y[2] - z[12,2] <= 1 R168: x[12] + y[3] - z[12,3] <= 1 R169: x[12] + y[4] - z[12,4] <= 1 R170: x[12] + y[5] - z[12,5] <= 1 R171: x[12] + y[6] - z[12,6] <= 1 R172: x[12] + y[7] - z[12,7] <= 1 R173: x[12] + y[8] - z[12,8] <= 1 R174: x[12] + y[9] - z[12,9] <= 1 R175: x[12] + y[10] - z[12,10] <= 1 R176: x[12] + y[11] - z[12,11] <= 1 R177: x[12] + y[12] - z[12,12] <= 1 R178: x[12] + y[13] - z[12,13] <= 1 R179: x[12] + y[14] - z[12,14] <= 1 R180: x[12] + y[15] - z[12,15] <= 1 R181: x[13] + y[1] - z[13,1] <= 1 R182: x[13] + y[2] - z[13,2] <= 1 R183: x[13] + y[3] - z[13,3] <= 1 R184: x[13] + y[4] - z[13,4] <= 1 R185: x[13] + y[5] - z[13,5] <= 1 R186: x[13] + y[6] - z[13,6] <= 1 R187: x[13] + y[7] - z[13,7] <= 1 R188: x[13] + y[8] - z[13,8] <= 1 R189: x[13] + y[9] - z[13,9] <= 1 R190: x[13] + y[10] - z[13,10] <= 1 R191: x[13] + y[11] - z[13,11] <= 1 R192: x[13] + y[12] - z[13,12] <= 1 R193: x[13] + y[13] - z[13,13] <= 1 R194: x[13] + y[14] - z[13,14] <= 1 R195: x[13] + y[15] - z[13,15] <= 1 R196: x[14] + y[1] - z[14,1] <= 1 R197: x[14] + y[2] - z[14,2] <= 1 R198: x[14] + y[3] - z[14,3] <= 1 R199: x[14] + y[4] - z[14,4] <= 1 R200: x[14] + y[5] - z[14,5] <= 1 R201: x[14] + y[6] - z[14,6] <= 1 R202: x[14] + y[7] - z[14,7] <= 1 R203: x[14] + y[8] - z[14,8] <= 1 R204: x[14] + y[9] - z[14,9] <= 1 R205: x[14] + y[10] - z[14,10] <= 1 R206: x[14] + y[11] - z[14,11] <= 1 R207: x[14] + y[12] - z[14,12] <= 1 R208: x[14] + y[13] - z[14,13] <= 1 R209: x[14] + y[14] - z[14,14] <= 1 R210: x[14] + y[15] - z[14,15] <= 1 R211: x[15] + y[1] - z[15,1] <= 1 R212: x[15] + y[2] - z[15,2] <= 1 R213: x[15] + y[3] - z[15,3] <= 1 R214: x[15] + y[4] - z[15,4] <= 1 R215: x[15] + y[5] - z[15,5] <= 1 R216: x[15] + y[6] - z[15,6] <= 1 R217: x[15] + y[7] - z[15,7] <= 1 R218: x[15] + y[8] - z[15,8] <= 1 R219: x[15] + y[9] - z[15,9] <= 1 R220: x[15] + y[10] - z[15,10] <= 1 R221: x[15] + y[11] - z[15,11] <= 1 R222: x[15] + y[12] - z[15,12] <= 1 R223: x[15] + y[13] - z[15,13] <= 1 R224: x[15] + y[14] - z[15,14] <= 1 R225: x[15] + y[15] - z[15,15] <= 1 R226: - x[1] + z[1,1] <= 0 R227: - x[1] + z[1,2] <= 0 R228: - x[1] + z[1,3] <= 0 R229: - x[1] + z[1,4] <= 0 R230: - x[1] + z[1,5] <= 0 R231: - x[1] + z[1,6] <= 0 R232: - x[1] + z[1,7] <= 0 R233: - x[1] + z[1,8] <= 0 R234: - x[1] + z[1,9] <= 0 R235: - x[1] + z[1,10] <= 0 R236: - x[1] + z[1,11] <= 0 R237: - x[1] + z[1,12] <= 0 R238: - x[1] + z[1,13] <= 0 R239: - x[1] + z[1,14] <= 0 R240: - x[1] + z[1,15] <= 0 R241: - x[2] + z[2,1] <= 0 R242: - x[2] + z[2,2] <= 0 R243: - x[2] + z[2,3] <= 0 R244: - x[2] + z[2,4] <= 0 R245: - x[2] + z[2,5] <= 0 R246: - x[2] + z[2,6] <= 0 R247: - x[2] + z[2,7] <= 0 R248: - x[2] + z[2,8] <= 0 R249: - x[2] + z[2,9] <= 0 R250: - x[2] + z[2,10] <= 0 R251: - x[2] + z[2,11] <= 0 R252: - x[2] + z[2,12] <= 0 R253: - x[2] + z[2,13] <= 0 R254: - x[2] + z[2,14] <= 0 R255: - x[2] + z[2,15] <= 0 R256: - x[3] + z[3,1] <= 0 R257: - x[3] + z[3,2] <= 0 R258: - x[3] + z[3,3] <= 0 R259: - x[3] + z[3,4] <= 0 R260: - x[3] + z[3,5] <= 0 R261: - x[3] + z[3,6] <= 0 R262: - x[3] + z[3,7] <= 0 R263: - x[3] + z[3,8] <= 0 R264: - x[3] + z[3,9] <= 0 R265: - x[3] + z[3,10] <= 0 R266: - x[3] + z[3,11] <= 0 R267: - x[3] + z[3,12] <= 0 R268: - x[3] + z[3,13] <= 0 R269: - x[3] + z[3,14] <= 0 R270: - x[3] + z[3,15] <= 0 R271: - x[4] + z[4,1] <= 0 R272: - x[4] + z[4,2] <= 0 R273: - x[4] + z[4,3] <= 0 R274: - x[4] + z[4,4] <= 0 R275: - x[4] + z[4,5] <= 0 R276: - x[4] + z[4,6] <= 0 R277: - x[4] + z[4,7] <= 0 R278: - x[4] + z[4,8] <= 0 R279: - x[4] + z[4,9] <= 0 R280: - x[4] + z[4,10] <= 0 R281: - x[4] + z[4,11] <= 0 R282: - x[4] + z[4,12] <= 0 R283: - x[4] + z[4,13] <= 0 R284: - x[4] + z[4,14] <= 0 R285: - x[4] + z[4,15] <= 0 R286: - x[5] + z[5,1] <= 0 R287: - x[5] + z[5,2] <= 0 R288: - x[5] + z[5,3] <= 0 R289: - x[5] + z[5,4] <= 0 R290: - x[5] + z[5,5] <= 0 R291: - x[5] + z[5,6] <= 0 R292: - x[5] + z[5,7] <= 0 R293: - x[5] + z[5,8] <= 0 R294: - x[5] + z[5,9] <= 0 R295: - x[5] + z[5,10] <= 0 R296: - x[5] + z[5,11] <= 0 R297: - x[5] + z[5,12] <= 0 R298: - x[5] + z[5,13] <= 0 R299: - x[5] + z[5,14] <= 0 R300: - x[5] + z[5,15] <= 0 R301: - x[6] + z[6,1] <= 0 R302: - x[6] + z[6,2] <= 0 R303: - x[6] + z[6,3] <= 0 R304: - x[6] + z[6,4] <= 0 R305: - x[6] + z[6,5] <= 0 R306: - x[6] + z[6,6] <= 0 R307: - x[6] + z[6,7] <= 0 R308: - x[6] + z[6,8] <= 0 R309: - x[6] + z[6,9] <= 0 R310: - x[6] + z[6,10] <= 0 R311: - x[6] + z[6,11] <= 0 R312: - x[6] + z[6,12] <= 0 R313: - x[6] + z[6,13] <= 0 R314: - x[6] + z[6,14] <= 0 R315: - x[6] + z[6,15] <= 0 R316: - x[7] + z[7,1] <= 0 R317: - x[7] + z[7,2] <= 0 R318: - x[7] + z[7,3] <= 0 R319: - x[7] + z[7,4] <= 0 R320: - x[7] + z[7,5] <= 0 R321: - x[7] + z[7,6] <= 0 R322: - x[7] + z[7,7] <= 0 R323: - x[7] + z[7,8] <= 0 R324: - x[7] + z[7,9] <= 0 R325: - x[7] + z[7,10] <= 0 R326: - x[7] + z[7,11] <= 0 R327: - x[7] + z[7,12] <= 0 R328: - x[7] + z[7,13] <= 0 R329: - x[7] + z[7,14] <= 0 R330: - x[7] + z[7,15] <= 0 R331: - x[8] + z[8,1] <= 0 R332: - x[8] + z[8,2] <= 0 R333: - x[8] + z[8,3] <= 0 R334: - x[8] + z[8,4] <= 0 R335: - x[8] + z[8,5] <= 0 R336: - x[8] + z[8,6] <= 0 R337: - x[8] + z[8,7] <= 0 R338: - x[8] + z[8,8] <= 0 R339: - x[8] + z[8,9] <= 0 R340: - x[8] + z[8,10] <= 0 R341: - x[8] + z[8,11] <= 0 R342: - x[8] + z[8,12] <= 0 R343: - x[8] + z[8,13] <= 0 R344: - x[8] + z[8,14] <= 0 R345: - x[8] + z[8,15] <= 0 R346: - x[9] + z[9,1] <= 0 R347: - x[9] + z[9,2] <= 0 R348: - x[9] + z[9,3] <= 0 R349: - x[9] + z[9,4] <= 0 R350: - x[9] + z[9,5] <= 0 R351: - x[9] + z[9,6] <= 0 R352: - x[9] + z[9,7] <= 0 R353: - x[9] + z[9,8] <= 0 R354: - x[9] + z[9,9] <= 0 R355: - x[9] + z[9,10] <= 0 R356: - x[9] + z[9,11] <= 0 R357: - x[9] + z[9,12] <= 0 R358: - x[9] + z[9,13] <= 0 R359: - x[9] + z[9,14] <= 0 R360: - x[9] + z[9,15] <= 0 R361: - x[10] + z[10,1] <= 0 R362: - x[10] + z[10,2] <= 0 R363: - x[10] + z[10,3] <= 0 R364: - x[10] + z[10,4] <= 0 R365: - x[10] + z[10,5] <= 0 R366: - x[10] + z[10,6] <= 0 R367: - x[10] + z[10,7] <= 0 R368: - x[10] + z[10,8] <= 0 R369: - x[10] + z[10,9] <= 0 R370: - x[10] + z[10,10] <= 0 R371: - x[10] + z[10,11] <= 0 R372: - x[10] + z[10,12] <= 0 R373: - x[10] + z[10,13] <= 0 R374: - x[10] + z[10,14] <= 0 R375: - x[10] + z[10,15] <= 0 R376: - x[11] + z[11,1] <= 0 R377: - x[11] + z[11,2] <= 0 R378: - x[11] + z[11,3] <= 0 R379: - x[11] + z[11,4] <= 0 R380: - x[11] + z[11,5] <= 0 R381: - x[11] + z[11,6] <= 0 R382: - x[11] + z[11,7] <= 0 R383: - x[11] + z[11,8] <= 0 R384: - x[11] + z[11,9] <= 0 R385: - x[11] + z[11,10] <= 0 R386: - x[11] + z[11,11] <= 0 R387: - x[11] + z[11,12] <= 0 R388: - x[11] + z[11,13] <= 0 R389: - x[11] + z[11,14] <= 0 R390: - x[11] + z[11,15] <= 0 R391: - x[12] + z[12,1] <= 0 R392: - x[12] + z[12,2] <= 0 R393: - x[12] + z[12,3] <= 0 R394: - x[12] + z[12,4] <= 0 R395: - x[12] + z[12,5] <= 0 R396: - x[12] + z[12,6] <= 0 R397: - x[12] + z[12,7] <= 0 R398: - x[12] + z[12,8] <= 0 R399: - x[12] + z[12,9] <= 0 R400: - x[12] + z[12,10] <= 0 R401: - x[12] + z[12,11] <= 0 R402: - x[12] + z[12,12] <= 0 R403: - x[12] + z[12,13] <= 0 R404: - x[12] + z[12,14] <= 0 R405: - x[12] + z[12,15] <= 0 R406: - x[13] + z[13,1] <= 0 R407: - x[13] + z[13,2] <= 0 R408: - x[13] + z[13,3] <= 0 R409: - x[13] + z[13,4] <= 0 R410: - x[13] + z[13,5] <= 0 R411: - x[13] + z[13,6] <= 0 R412: - x[13] + z[13,7] <= 0 R413: - x[13] + z[13,8] <= 0 R414: - x[13] + z[13,9] <= 0 R415: - x[13] + z[13,10] <= 0 R416: - x[13] + z[13,11] <= 0 R417: - x[13] + z[13,12] <= 0 R418: - x[13] + z[13,13] <= 0 R419: - x[13] + z[13,14] <= 0 R420: - x[13] + z[13,15] <= 0 R421: - x[14] + z[14,1] <= 0 R422: - x[14] + z[14,2] <= 0 R423: - x[14] + z[14,3] <= 0 R424: - x[14] + z[14,4] <= 0 R425: - x[14] + z[14,5] <= 0 R426: - x[14] + z[14,6] <= 0 R427: - x[14] + z[14,7] <= 0 R428: - x[14] + z[14,8] <= 0 R429: - x[14] + z[14,9] <= 0 R430: - x[14] + z[14,10] <= 0 R431: - x[14] + z[14,11] <= 0 R432: - x[14] + z[14,12] <= 0 R433: - x[14] + z[14,13] <= 0 R434: - x[14] + z[14,14] <= 0 R435: - x[14] + z[14,15] <= 0 R436: - x[15] + z[15,1] <= 0 R437: - x[15] + z[15,2] <= 0 R438: - x[15] + z[15,3] <= 0 R439: - x[15] + z[15,4] <= 0 R440: - x[15] + z[15,5] <= 0 R441: - x[15] + z[15,6] <= 0 R442: - x[15] + z[15,7] <= 0 R443: - x[15] + z[15,8] <= 0 R444: - x[15] + z[15,9] <= 0 R445: - x[15] + z[15,10] <= 0 R446: - x[15] + z[15,11] <= 0 R447: - x[15] + z[15,12] <= 0 R448: - x[15] + z[15,13] <= 0 R449: - x[15] + z[15,14] <= 0 R450: - x[15] + z[15,15] <= 0 R451: - y[1] + z[1,1] <= 0 R452: - y[2] + z[1,2] <= 0 R453: - y[3] + z[1,3] <= 0 R454: - y[4] + z[1,4] <= 0 R455: - y[5] + z[1,5] <= 0 R456: - y[6] + z[1,6] <= 0 R457: - y[7] + z[1,7] <= 0 R458: - y[8] + z[1,8] <= 0 R459: - y[9] + z[1,9] <= 0 R460: - y[10] + z[1,10] <= 0 R461: - y[11] + z[1,11] <= 0 R462: - y[12] + z[1,12] <= 0 R463: - y[13] + z[1,13] <= 0 R464: - y[14] + z[1,14] <= 0 R465: - y[15] + z[1,15] <= 0 R466: - y[1] + z[2,1] <= 0 R467: - y[2] + z[2,2] <= 0 R468: - y[3] + z[2,3] <= 0 R469: - y[4] + z[2,4] <= 0 R470: - y[5] + z[2,5] <= 0 R471: - y[6] + z[2,6] <= 0 R472: - y[7] + z[2,7] <= 0 R473: - y[8] + z[2,8] <= 0 R474: - y[9] + z[2,9] <= 0 R475: - y[10] + z[2,10] <= 0 R476: - y[11] + z[2,11] <= 0 R477: - y[12] + z[2,12] <= 0 R478: - y[13] + z[2,13] <= 0 R479: - y[14] + z[2,14] <= 0 R480: - y[15] + z[2,15] <= 0 R481: - y[1] + z[3,1] <= 0 R482: - y[2] + z[3,2] <= 0 R483: - y[3] + z[3,3] <= 0 R484: - y[4] + z[3,4] <= 0 R485: - y[5] + z[3,5] <= 0 R486: - y[6] + z[3,6] <= 0 R487: - y[7] + z[3,7] <= 0 R488: - y[8] + z[3,8] <= 0 R489: - y[9] + z[3,9] <= 0 R490: - y[10] + z[3,10] <= 0 R491: - y[11] + z[3,11] <= 0 R492: - y[12] + z[3,12] <= 0 R493: - y[13] + z[3,13] <= 0 R494: - y[14] + z[3,14] <= 0 R495: - y[15] + z[3,15] <= 0 R496: - y[1] + z[4,1] <= 0 R497: - y[2] + z[4,2] <= 0 R498: - y[3] + z[4,3] <= 0 R499: - y[4] + z[4,4] <= 0 R500: - y[5] + z[4,5] <= 0 R501: - y[6] + z[4,6] <= 0 R502: - y[7] + z[4,7] <= 0 R503: - y[8] + z[4,8] <= 0 R504: - y[9] + z[4,9] <= 0 R505: - y[10] + z[4,10] <= 0 R506: - y[11] + z[4,11] <= 0 R507: - y[12] + z[4,12] <= 0 R508: - y[13] + z[4,13] <= 0 R509: - y[14] + z[4,14] <= 0 R510: - y[15] + z[4,15] <= 0 R511: - y[1] + z[5,1] <= 0 R512: - y[2] + z[5,2] <= 0 R513: - y[3] + z[5,3] <= 0 R514: - y[4] + z[5,4] <= 0 R515: - y[5] + z[5,5] <= 0 R516: - y[6] + z[5,6] <= 0 R517: - y[7] + z[5,7] <= 0 R518: - y[8] + z[5,8] <= 0 R519: - y[9] + z[5,9] <= 0 R520: - y[10] + z[5,10] <= 0 R521: - y[11] + z[5,11] <= 0 R522: - y[12] + z[5,12] <= 0 R523: - y[13] + z[5,13] <= 0 R524: - y[14] + z[5,14] <= 0 R525: - y[15] + z[5,15] <= 0 R526: - y[1] + z[6,1] <= 0 R527: - y[2] + z[6,2] <= 0 R528: - y[3] + z[6,3] <= 0 R529: - y[4] + z[6,4] <= 0 R530: - y[5] + z[6,5] <= 0 R531: - y[6] + z[6,6] <= 0 R532: - y[7] + z[6,7] <= 0 R533: - y[8] + z[6,8] <= 0 R534: - y[9] + z[6,9] <= 0 R535: - y[10] + z[6,10] <= 0 R536: - y[11] + z[6,11] <= 0 R537: - y[12] + z[6,12] <= 0 R538: - y[13] + z[6,13] <= 0 R539: - y[14] + z[6,14] <= 0 R540: - y[15] + z[6,15] <= 0 R541: - y[1] + z[7,1] <= 0 R542: - y[2] + z[7,2] <= 0 R543: - y[3] + z[7,3] <= 0 R544: - y[4] + z[7,4] <= 0 R545: - y[5] + z[7,5] <= 0 R546: - y[6] + z[7,6] <= 0 R547: - y[7] + z[7,7] <= 0 R548: - y[8] + z[7,8] <= 0 R549: - y[9] + z[7,9] <= 0 R550: - y[10] + z[7,10] <= 0 R551: - y[11] + z[7,11] <= 0 R552: - y[12] + z[7,12] <= 0 R553: - y[13] + z[7,13] <= 0 R554: - y[14] + z[7,14] <= 0 R555: - y[15] + z[7,15] <= 0 R556: - y[1] + z[8,1] <= 0 R557: - y[2] + z[8,2] <= 0 R558: - y[3] + z[8,3] <= 0 R559: - y[4] + z[8,4] <= 0 R560: - y[5] + z[8,5] <= 0 R561: - y[6] + z[8,6] <= 0 R562: - y[7] + z[8,7] <= 0 R563: - y[8] + z[8,8] <= 0 R564: - y[9] + z[8,9] <= 0 R565: - y[10] + z[8,10] <= 0 R566: - y[11] + z[8,11] <= 0 R567: - y[12] + z[8,12] <= 0 R568: - y[13] + z[8,13] <= 0 R569: - y[14] + z[8,14] <= 0 R570: - y[15] + z[8,15] <= 0 R571: - y[1] + z[9,1] <= 0 R572: - y[2] + z[9,2] <= 0 R573: - y[3] + z[9,3] <= 0 R574: - y[4] + z[9,4] <= 0 R575: - y[5] + z[9,5] <= 0 R576: - y[6] + z[9,6] <= 0 R577: - y[7] + z[9,7] <= 0 R578: - y[8] + z[9,8] <= 0 R579: - y[9] + z[9,9] <= 0 R580: - y[10] + z[9,10] <= 0 R581: - y[11] + z[9,11] <= 0 R582: - y[12] + z[9,12] <= 0 R583: - y[13] + z[9,13] <= 0 R584: - y[14] + z[9,14] <= 0 R585: - y[15] + z[9,15] <= 0 R586: - y[1] + z[10,1] <= 0 R587: - y[2] + z[10,2] <= 0 R588: - y[3] + z[10,3] <= 0 R589: - y[4] + z[10,4] <= 0 R590: - y[5] + z[10,5] <= 0 R591: - y[6] + z[10,6] <= 0 R592: - y[7] + z[10,7] <= 0 R593: - y[8] + z[10,8] <= 0 R594: - y[9] + z[10,9] <= 0 R595: - y[10] + z[10,10] <= 0 R596: - y[11] + z[10,11] <= 0 R597: - y[12] + z[10,12] <= 0 R598: - y[13] + z[10,13] <= 0 R599: - y[14] + z[10,14] <= 0 R600: - y[15] + z[10,15] <= 0 R601: - y[1] + z[11,1] <= 0 R602: - y[2] + z[11,2] <= 0 R603: - y[3] + z[11,3] <= 0 R604: - y[4] + z[11,4] <= 0 R605: - y[5] + z[11,5] <= 0 R606: - y[6] + z[11,6] <= 0 R607: - y[7] + z[11,7] <= 0 R608: - y[8] + z[11,8] <= 0 R609: - y[9] + z[11,9] <= 0 R610: - y[10] + z[11,10] <= 0 R611: - y[11] + z[11,11] <= 0 R612: - y[12] + z[11,12] <= 0 R613: - y[13] + z[11,13] <= 0 R614: - y[14] + z[11,14] <= 0 R615: - y[15] + z[11,15] <= 0 R616: - y[1] + z[12,1] <= 0 R617: - y[2] + z[12,2] <= 0 R618: - y[3] + z[12,3] <= 0 R619: - y[4] + z[12,4] <= 0 R620: - y[5] + z[12,5] <= 0 R621: - y[6] + z[12,6] <= 0 R622: - y[7] + z[12,7] <= 0 R623: - y[8] + z[12,8] <= 0 R624: - y[9] + z[12,9] <= 0 R625: - y[10] + z[12,10] <= 0 R626: - y[11] + z[12,11] <= 0 R627: - y[12] + z[12,12] <= 0 R628: - y[13] + z[12,13] <= 0 R629: - y[14] + z[12,14] <= 0 R630: - y[15] + z[12,15] <= 0 R631: - y[1] + z[13,1] <= 0 R632: - y[2] + z[13,2] <= 0 R633: - y[3] + z[13,3] <= 0 R634: - y[4] + z[13,4] <= 0 R635: - y[5] + z[13,5] <= 0 R636: - y[6] + z[13,6] <= 0 R637: - y[7] + z[13,7] <= 0 R638: - y[8] + z[13,8] <= 0 R639: - y[9] + z[13,9] <= 0 R640: - y[10] + z[13,10] <= 0 R641: - y[11] + z[13,11] <= 0 R642: - y[12] + z[13,12] <= 0 R643: - y[13] + z[13,13] <= 0 R644: - y[14] + z[13,14] <= 0 R645: - y[15] + z[13,15] <= 0 R646: - y[1] + z[14,1] <= 0 R647: - y[2] + z[14,2] <= 0 R648: - y[3] + z[14,3] <= 0 R649: - y[4] + z[14,4] <= 0 R650: - y[5] + z[14,5] <= 0 R651: - y[6] + z[14,6] <= 0 R652: - y[7] + z[14,7] <= 0 R653: - y[8] + z[14,8] <= 0 R654: - y[9] + z[14,9] <= 0 R655: - y[10] + z[14,10] <= 0 R656: - y[11] + z[14,11] <= 0 R657: - y[12] + z[14,12] <= 0 R658: - y[13] + z[14,13] <= 0 R659: - y[14] + z[14,14] <= 0 R660: - y[15] + z[14,15] <= 0 R661: - y[1] + z[15,1] <= 0 R662: - y[2] + z[15,2] <= 0 R663: - y[3] + z[15,3] <= 0 R664: - y[4] + z[15,4] <= 0 R665: - y[5] + z[15,5] <= 0 R666: - y[6] + z[15,6] <= 0 R667: - y[7] + z[15,7] <= 0 R668: - y[8] + z[15,8] <= 0 R669: - y[9] + z[15,9] <= 0 R670: - y[10] + z[15,10] <= 0 R671: - y[11] + z[15,11] <= 0 R672: - y[12] + z[15,12] <= 0 R673: - y[13] + z[15,13] <= 0 R674: - y[14] + z[15,14] <= 0 R675: - y[15] + z[15,15] <= 0Bounds x[1] <= 1 x[2] <= 1 x[3] <= 1 x[4] <= 1 x[5] <= 1 x[6] <= 1 x[7] <= 1 x[8] <= 1 x[9] <= 1 x[10] <= 1 x[11] <= 1 x[12] <= 1 x[13] <= 1 x[14] <= 1 x[15] <= 1 y[1] <= 1 y[2] <= 1 y[3] <= 1 y[4] <= 1 y[5] <= 1 y[6] <= 1 y[7] <= 1 y[8] <= 1 y[9] <= 1 y[10] <= 1 y[11] <= 1 y[12] <= 1 y[13] <= 1 y[14] <= 1 y[15] <= 1 z[1,1] <= 1 z[1,2] <= 1 z[1,3] <= 1 z[1,4] <= 1 z[1,5] <= 1 z[1,6] <= 1 z[1,7] <= 1 z[1,8] <= 1 z[1,9] <= 1 z[1,10] <= 1 z[1,11] <= 1 z[1,12] <= 1 z[1,13] <= 1 z[1,14] <= 1 z[1,15] <= 1 z[2,1] <= 1 z[2,2] <= 1 z[2,3] <= 1 z[2,4] <= 1 z[2,5] <= 1 z[2,6] <= 1 z[2,7] <= 1 z[2,8] <= 1 z[2,9] <= 1 z[2,10] <= 1 z[2,11] <= 1 z[2,12] <= 1 z[2,13] <= 1 z[2,14] <= 1 z[2,15] <= 1 z[3,1] <= 1 z[3,2] <= 1 z[3,3] <= 1 z[3,4] <= 1 z[3,5] <= 1 z[3,6] <= 1 z[3,7] <= 1 z[3,8] <= 1 z[3,9] <= 1 z[3,10] <= 1 z[3,11] <= 1 z[3,12] <= 1 z[3,13] <= 1 z[3,14] <= 1 z[3,15] <= 1 z[4,1] <= 1 z[4,2] <= 1 z[4,3] <= 1 z[4,4] <= 1 z[4,5] <= 1 z[4,6] <= 1 z[4,7] <= 1 z[4,8] <= 1 z[4,9] <= 1 z[4,10] <= 1 z[4,11] <= 1 z[4,12] <= 1 z[4,13] <= 1 z[4,14] <= 1 z[4,15] <= 1 z[5,1] <= 1 z[5,2] <= 1 z[5,3] <= 1 z[5,4] <= 1 z[5,5] <= 1 z[5,6] <= 1 z[5,7] <= 1 z[5,8] <= 1 z[5,9] <= 1 z[5,10] <= 1 z[5,11] <= 1 z[5,12] <= 1 z[5,13] <= 1 z[5,14] <= 1 z[5,15] <= 1 z[6,1] <= 1 z[6,2] <= 1 z[6,3] <= 1 z[6,4] <= 1 z[6,5] <= 1 z[6,6] <= 1 z[6,7] <= 1 z[6,8] <= 1 z[6,9] <= 1 z[6,10] <= 1 z[6,11] <= 1 z[6,12] <= 1 z[6,13] <= 1 z[6,14] <= 1 z[6,15] <= 1 z[7,1] <= 1 z[7,2] <= 1 z[7,3] <= 1 z[7,4] <= 1 z[7,5] <= 1 z[7,6] <= 1 z[7,7] <= 1 z[7,8] <= 1 z[7,9] <= 1 z[7,10] <= 1 z[7,11] <= 1 z[7,12] <= 1 z[7,13] <= 1 z[7,14] <= 1 z[7,15] <= 1 z[8,1] <= 1 z[8,2] <= 1 z[8,3] <= 1 z[8,4] <= 1 z[8,5] <= 1 z[8,6] <= 1 z[8,7] <= 1 z[8,8] <= 1 z[8,9] <= 1 z[8,10] <= 1 z[8,11] <= 1 z[8,12] <= 1 z[8,13] <= 1 z[8,14] <= 1 z[8,15] <= 1 z[9,1] <= 1 z[9,2] <= 1 z[9,3] <= 1 z[9,4] <= 1 z[9,5] <= 1 z[9,6] <= 1 z[9,7] <= 1 z[9,8] <= 1 z[9,9] <= 1 z[9,10] <= 1 z[9,11] <= 1 z[9,12] <= 1 z[9,13] <= 1 z[9,14] <= 1 z[9,15] <= 1 z[10,1] <= 1 z[10,2] <= 1 z[10,3] <= 1 z[10,4] <= 1 z[10,5] <= 1 z[10,6] <= 1 z[10,7] <= 1 z[10,8] <= 1 z[10,9] <= 1 z[10,10] <= 1 z[10,11] <= 1 z[10,12] <= 1 z[10,13] <= 1 z[10,14] <= 1 z[10,15] <= 1 z[11,1] <= 1 z[11,2] <= 1 z[11,3] <= 1 z[11,4] <= 1 z[11,5] <= 1 z[11,6] <= 1 z[11,7] <= 1 z[11,8] <= 1 z[11,9] <= 1 z[11,10] <= 1 z[11,11] <= 1 z[11,12] <= 1 z[11,13] <= 1 z[11,14] <= 1 z[11,15] <= 1 z[12,1] <= 1 z[12,2] <= 1 z[12,3] <= 1 z[12,4] <= 1 z[12,5] <= 1 z[12,6] <= 1 z[12,7] <= 1 z[12,8] <= 1 z[12,9] <= 1 z[12,10] <= 1 z[12,11] <= 1 z[12,12] <= 1 z[12,13] <= 1 z[12,14] <= 1 z[12,15] <= 1 z[13,1] <= 1 z[13,2] <= 1 z[13,3] <= 1 z[13,4] <= 1 z[13,5] <= 1 z[13,6] <= 1 z[13,7] <= 1 z[13,8] <= 1 z[13,9] <= 1 z[13,10] <= 1 z[13,11] <= 1 z[13,12] <= 1 z[13,13] <= 1 z[13,14] <= 1 z[13,15] <= 1 z[14,1] <= 1 z[14,2] <= 1 z[14,3] <= 1 z[14,4] <= 1 z[14,5] <= 1 z[14,6] <= 1 z[14,7] <= 1 z[14,8] <= 1 z[14,9] <= 1 z[14,10] <= 1 z[14,11] <= 1 z[14,12] <= 1 z[14,13] <= 1 z[14,14] <= 1 z[14,15] <= 1 z[15,1] <= 1 z[15,2] <= 1 z[15,3] <= 1 z[15,4] <= 1 z[15,5] <= 1 z[15,6] <= 1 z[15,7] <= 1 z[15,8] <= 1 z[15,9] <= 1 z[15,10] <= 1 z[15,11] <= 1 z[15,12] <= 1 z[15,13] <= 1 z[15,14] <= 1 z[15,15] <= 1End
• Gurobi Staff

Yes, there is something still happening. I will let you know once I found out.

Thanks

• Gurobi Staff

Hi Jan,

I double checked the behavior and what is happening is indeed expected. In any node (and also the root node), Gurobi performs trivial bound propagations irrespective of the Presolve setting. In your particular case, this means that the linear constraint

R0: x[1] + x[2] + x[3] <= 0.48

is used to tighten the upper bounds of variables $$x_1, x_2, x_3$$ to $$0.48$$. This explains the difference in the objective value between your manual McCormick relaxation and the root relaxation bound reported by Gurobi.

If you would like to only check whether your McCormick relaxation is correct, you could remove all non-bilinear constraints from your model. This would prevent Gurobi from taking advantage from any trivially deducible bound information.

I hope this helps.

Best regards,
Jaromił

yes, this helps. Thank you very much!

Is there a special reason, why these bound propagations are not considered as Presolve? In my opinion, it is unnecessarily confusing to the user that sets Presolve=0.

Best,
Jan

• Gurobi Staff

Hi Jan,

Is there a special reason, why these bound propagations are not considered as Presolve? In my opinion, it is unnecessarily confusing to the user that sets Presolve=0.

Presolve controls what happens before the solution process (thus the name).

Bound propagation for particular nodes are performed outside of presolve and thus not affected by it. We discussed this issue and came to the conclusion that it does not make sense for the average user to have control over bound propagation within nodes. Of course there are special cases like yours, but for that we can provide a workaround as discussed above.

Best regards,
Jaromił

Hello Jaromił Najman,

okay, thanks for that explanation. :)

Best,

Jan