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
  C0 + C1 + C2 + C3
Subject To
 R0: C5 = 1
 R1: C6 = 1
 R2: C7 = 1
 R3: C8 = 1
 R4: - C0 - C4 <= 0
 R5: - C1 - C4 <= 0
 R6: - C2 - C4 <= 0
 R7: - C3 - C4 <= 0
 R8: - C5 - C9 <= 0
 R9: - C6 - C9 <= 0
 R10: - C7 - C9 <= 0
 R11: - C8 - C9 <= 0
 R12: C4 + C9 <= -1
Bounds
 C0 free
 C1 free
 C2 free
 C3 free
 C4 free
 C5 free
 C6 free
 C7 free
 C8 free
 C9 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