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
  C0 + C1 + C2 + C3
Subject To
 R0: - C0 + C4 + C8 <= 0
 R1: - C1 + C5 + C9 <= 0
 R2: - C2 + C6 + C10 <= 0
 R3: - C3 + C7 + C11 <= 0
 R4: - C0 - C4 - C8 <= 0
 R5: - C1 - C5 - C9 <= 0
 R6: - C2 - C6 - C10 <= 0
 R7: - C3 - C7 - C11 <= 0
 R8: - C4 <= 0
 R9: - C5 <= 0
 R10: - C6 <= 0
 R11: - C7 <= 0
 R12: - C8 <= 0
 R13: - C9 <= 0
 R14: - C10 <= 0
 R15: - C11 <= 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
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