CVXPY
Minimize
  Sum(abs(X + Y), None, False)
Subject To
Bounds
 0.0 <= X
 0.0 <= Y
End
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AFTER COMPILATION
Minimize
  x_0 + x_1 + x_2 + x_3
Subject To
 R0: - x_0 + x_4 + x_8 <= 0
 R1: - x_1 + x_5 + x_9 <= 0
 R2: - x_2 + x_6 + x_10 <= 0
 R3: - x_3 + x_7 + x_11 <= 0
 R4: - x_0 - x_4 - x_8 <= 0
 R5: - x_1 - x_5 - x_9 <= 0
 R6: - x_2 - x_6 - x_10 <= 0
 R7: - x_3 - x_7 - x_11 <= 0
Bounds
 x_0 free
 x_1 free
 x_2 free
 x_3 free
End
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GUROBI
Minimize
  abs_2 + abs_4 + abs_6 + abs_8
Subject To
 R0: - X[0,0] - Y[0,0] + index_1 = 0
 R1: - X[0,1] - Y[0,1] + index_3 = 0
 R2: - X[1,0] - Y[1,0] + index_5 = 0
 R3: - X[1,1] - Y[1,1] + index_7 = 0
Bounds
 index_1 free
 index_3 free
 index_5 free
 index_7 free
General Constraints
 GC0: abs_2 = ABS ( index_1 )
 GC1: abs_4 = ABS ( index_3 )
 GC2: abs_6 = ABS ( index_5 )
 GC3: abs_8 = ABS ( index_7 )
End