CVXPY
Minimize
  Sum(X + Y, None, False)
Subject To
 23: [[1.00 -2.00]
 [3.00 4.00]] <= minimum(X, Y)
Bounds
 X free
 Y free
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: - C4 - C8 <= 0
 R5: - C5 - C9 <= 0
 R6: - C6 - C10 <= 0
 R7: - C7 - C11 <= 0
 R8: C8 <= -1
 R9: C9 <= -3
 R10: C10 <= 2
 R11: C11 <= -4
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
  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
 23[0,0]: minimum_1 >= 1
 23[0,1]: minimum_2 >= -2
 23[1,0]: minimum_3 >= 3
 23[1,1]: minimum_4 >= 4
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] )
 GC1: minimum_2 = MIN ( X[0,1] , Y[0,1] )
 GC2: minimum_3 = MIN ( X[1,0] , Y[1,0] )
 GC3: minimum_4 = MIN ( X[1,1] , Y[1,1] )
End