CVXPY
Minimize
  Sum(X, None, False)
Subject To
 40: 1.0 <= min(X + Y, None, False)
 45: Y == 1.0
Bounds
 X free
 Y free
End
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AFTER COMPILATION
Minimize
  C0 + C1 + C2 + C3
Subject To
 R0: C4 = 1
 R1: C5 = 1
 R2: C6 = 1
 R3: C7 = 1
 R4: - C0 - C4 - C8 <= 0
 R5: - C1 - C5 - C8 <= 0
 R6: - C2 - C6 - C8 <= 0
 R7: - C3 - C7 - C8 <= 0
 R8: C8 <= -1
Bounds
 C0 free
 C1 free
 C2 free
 C3 free
 C4 free
 C5 free
 C6 free
 C7 free
 C8 free
End
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GUROBI
Minimize
  X[0,0] + X[0,1] + X[1,0] + X[1,1]
Subject To
 R0: - X[0,0] - Y[0,0] + AddExpression_1[0,0] = 0
 R1: - X[0,1] - Y[0,1] + AddExpression_1[0,1] = 0
 R2: - X[1,0] - Y[1,0] + AddExpression_1[1,0] = 0
 R3: - X[1,1] - Y[1,1] + AddExpression_1[1,1] = 0
 40: min_2 >= 1
 45[0,0]: Y[0,0] = 1
 45[0,1]: Y[0,1] = 1
 45[1,0]: Y[1,0] = 1
 45[1,1]: Y[1,1] = 1
Bounds
 X[0,0] free
 X[0,1] free
 X[1,0] free
 X[1,1] free
 Y[0,0] free
 Y[0,1] free
 Y[1,0] free
 Y[1,1] free
 AddExpression_1[0,0] free
 AddExpression_1[0,1] free
 AddExpression_1[1,0] free
 AddExpression_1[1,1] free
 min_2 free
General Constraints
 GC0: min_2 = MIN ( AddExpression_1[0,0] , AddExpression_1[0,1] ,
   AddExpression_1[1,0] , AddExpression_1[1,1] )
End