CVXPY
Minimize
  Sum(X, None, False)
Subject To
 38: -Y <= minimum(X, [[1.00 -2.00]
 [3.00 4.00]])
 43: Y == 2.0
Bounds
 X free
 Y free
End
----------------------------------------
AFTER COMPILATION
Minimize
  C0 + C1 + C2 + C3
Subject To
 R0: C8 = 2
 R1: C9 = 2
 R2: C10 = 2
 R3: C11 = 2
 R4: - C0 - C4 <= 0
 R5: - C1 - C5 <= 0
 R6: - C2 - C6 <= 0
 R7: - C3 - C7 <= 0
 R8: - C4 <= 1
 R9: - C5 <= 3
 R10: - C6 <= -2
 R11: - C7 <= 4
 R12: C4 - C8 <= 0
 R13: C5 - C9 <= 0
 R14: C6 - C10 <= 0
 R15: C7 - 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
----------------------------------------
GUROBI
Minimize
  X[0,0] + X[0,1] + X[1,0] + X[1,1]
Subject To
 38[0,0]: - Y[0,0] - minimum_1 <= 0
 38[0,1]: - Y[0,1] - minimum_2 <= 0
 38[1,0]: - Y[1,0] - minimum_3 <= 0
 38[1,1]: - Y[1,1] - minimum_4 <= 0
 43[0,0]: Y[0,0] = 2
 43[0,1]: Y[0,1] = 2
 43[1,0]: Y[1,0] = 2
 43[1,1]: Y[1,1] = 2
Bounds
 X[0,0] free
 X[0,1] free
 X[1,0] free
 X[1,1] free
 Y[0,0] free
 Y[0,1] free
 Y[1,0] free
 Y[1,1] free
 minimum_1 free
 minimum_2 free
 minimum_3 free
 minimum_4 free
General Constraints
 GC0: minimum_1 = MIN ( X[0,0] , 1 )
 GC1: minimum_2 = MIN ( X[0,1] , -2 )
 GC2: minimum_3 = MIN ( X[1,0] , 3 )
 GC3: minimum_4 = MIN ( X[1,1] , 4 )
End