CVXPY
Minimize
  Sum(X + Y, None, False)
Subject To
 31: -2.0 <= minimum(X, Y, [[1.00 -2.00]
 [3.00 4.00]])
Bounds
 X free
 Y free
End
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AFTER COMPILATION
Minimize
  x_0 + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7
Subject To
 R0: - x_0 - x_8 <= 0
 R1: - x_1 - x_9 <= 0
 R2: - x_2 - x_10 <= 0
 R3: - x_3 - x_11 <= 0
 R4: - x_4 - x_8 <= 0
 R5: - x_5 - x_9 <= 0
 R6: - x_6 - x_10 <= 0
 R7: - x_7 - x_11 <= 0
 R8: - x_8 <= 1
 R9: - x_9 <= 3
 R10: - x_10 <= -2
 R11: - x_11 <= 4
 R12: x_8 <= 2
 R13: x_9 <= 2
 R14: x_10 <= 2
 R15: x_11 <= 2
Bounds
 x_0 free
 x_1 free
 x_2 free
 x_3 free
 x_4 free
 x_5 free
 x_6 free
 x_7 free
 x_8 free
 x_9 free
 x_10 free
 x_11 free
End
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GUROBI
Minimize
  X[0,0] + X[0,1] + X[1,0] + X[1,1] + Y[0,0] + Y[0,1] + Y[1,0] + Y[1,1]
Subject To
 31[0,0]: minimum_1 >= -2
 31[0,1]: minimum_2 >= -2
 31[1,0]: minimum_3 >= -2
 31[1,1]: minimum_4 >= -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] , Y[0,0] , 1 )
 GC1: minimum_2 = MIN ( X[0,1] , Y[0,1] , -2 )
 GC2: minimum_3 = MIN ( X[1,0] , Y[1,0] , 3 )
 GC3: minimum_4 = MIN ( X[1,1] , Y[1,1] , 4 )
End