CVXPY
Minimize
  Sum(X, None, False)
Subject To
 28: 1.0 <= min(X, None, False) + min(Y, None, False)
 33: Y == 1.0
Bounds
 X free
 Y free
End
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AFTER COMPILATION
Minimize
  x_0 + x_1 + x_2 + x_3
Subject To
 R0: x_5 = 1
 R1: x_6 = 1
 R2: x_7 = 1
 R3: x_8 = 1
 R4: - x_0 - x_4 <= 0
 R5: - x_1 - x_4 <= 0
 R6: - x_2 - x_4 <= 0
 R7: - x_3 - x_4 <= 0
 R8: - x_5 - x_9 <= 0
 R9: - x_6 - x_9 <= 0
 R10: - x_7 - x_9 <= 0
 R11: - x_8 - x_9 <= 0
 R12: x_4 + x_9 <= -1
Bounds
 x_0 free
 x_1 free
 x_2 free
 x_3 free
 x_4 free
 x_5 free
 x_6 free
 x_7 free
 x_8 free
 x_9 free
End
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GUROBI
Minimize
  X[0,0] + X[0,1] + X[1,0] + X[1,1]
Subject To
 28: min_1 + min_2 >= 1
 33[0,0]: Y[0,0] = 1
 33[0,1]: Y[0,1] = 1
 33[1,0]: Y[1,0] = 1
 33[1,1]: Y[1,1] = 1
Bounds
 X[0,0] free
 X[0,1] free
 X[1,0] free
 X[1,1] free
 min_1 free
 Y[0,0] free
 Y[0,1] free
 Y[1,0] free
 Y[1,1] free
 min_2 free
General Constraints
 GC0: min_1 = MIN ( X[0,0] , X[0,1] , X[1,0] , X[1,1] )
 GC1: min_2 = MIN ( Y[0,0] , Y[0,1] , Y[1,0] , Y[1,1] )
End