CVXPY
Minimize
  Sum(abs(X) + abs(Y), None, False)
Subject To
Bounds
 0.0 <= X
 0.0 <= Y
End
----------------------------------------
AFTER COMPILATION
Minimize
  C0 + C1 + C2 + C3 + C4 + C5 + C6 + C7
Subject To
 R0: - C0 + C8 <= 0
 R1: - C1 + C9 <= 0
 R2: - C2 + C10 <= 0
 R3: - C3 + C11 <= 0
 R4: - C0 - C8 <= 0
 R5: - C1 - C9 <= 0
 R6: - C2 - C10 <= 0
 R7: - C3 - C11 <= 0
 R8: - C4 + C12 <= 0
 R9: - C5 + C13 <= 0
 R10: - C6 + C14 <= 0
 R11: - C7 + C15 <= 0
 R12: - C4 - C12 <= 0
 R13: - C5 - C13 <= 0
 R14: - C6 - C14 <= 0
 R15: - C7 - C15 <= 0
Bounds
 C0 free
 C1 free
 C2 free
 C3 free
 C4 free
 C5 free
 C6 free
 C7 free
End
----------------------------------------
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
   + 0 Y[0,0] + 0 Y[0,1] + 0 Y[1,0] + 0 Y[1,1] + abs_5 + abs_6 + abs_7
   + abs_8
Subject To
Bounds
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] )
 GC4: abs_5 = ABS ( Y[0,0] )
 GC5: abs_6 = ABS ( Y[0,1] )
 GC6: abs_7 = ABS ( Y[1,0] )
 GC7: abs_8 = ABS ( Y[1,1] )
End