CVXPY
Minimize
  Sum(abs(X) + abs(Y), None, False)
Subject To
Bounds
 0.0 <= X
 0.0 <= Y
End
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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
 R16: - C8 <= 0
 R17: - C9 <= 0
 R18: - C10 <= 0
 R19: - C11 <= 0
 R20: - C12 <= 0
 R21: - C13 <= 0
 R22: - C14 <= 0
 R23: - C15 <= 0
Bounds
 C0 free
 C1 free
 C2 free
 C3 free
 C4 free
 C5 free
 C6 free
 C7 free
 C8 free
 C9 free
 C10 free
 C11 free
 C12 free
 C13 free
 C14 free
 C15 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
   + 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