CVXPY
Minimize
  Sum(abs(X) + abs([[1.00 -2.00]
 [3.00 4.00]]), None, False)
Subject To
Bounds
 0.0 <= X
End
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AFTER COMPILATION
Minimize
  C0 + C1 + C2 + C3
Subject To
 R0: - C0 + C4 <= 0
 R1: - C1 + C5 <= 0
 R2: - C2 + C6 <= 0
 R3: - C3 + C7 <= 0
 R4: - C0 - C4 <= 0
 R5: - C1 - C5 <= 0
 R6: - C2 - C6 <= 0
 R7: - C3 - C7 <= 0
 R8: - C4 <= 0
 R9: - C5 <= 0
 R10: - C6 <= 0
 R11: - C7 <= 0
Bounds
 C0 free
 C1 free
 C2 free
 C3 free
 C4 free
 C5 free
 C6 free
 C7 free
End
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GUROBI
Minimize
  0 X[0,0] + 0 X[0,1] + 0 X[1,0] + 0 X[1,1] + abs_1 + abs_2 + abs_3 + abs_4
   + 10 Constant
Subject To
Bounds
 Constant = 1
General Constraints
 GC0: abs_1 = ABS ( X[0,0] )
 GC1: abs_2 = ABS ( X[0,1] )
 GC2: abs_3 = ABS ( X[1,0] )
 GC3: abs_4 = ABS ( X[1,1] )
End