Absolute values

y = | x₁ - x₂| for two variables x₁, x₂ with 0 ≤ x_i ≤ U

• Introduce binary variables d₁, d₂ to mean

  d1 : 1 if x1 - x2 is the positive value

  d2 : 1 if x2 - x1 is the positive value

• MIP formulation

  0 ≤ xi ≤ U	[1.i]
  0 ≤ y - (x1-x2) ≤ 2 · U · d2	[2]
  0 ≤ y - (x2-x1) ≤ 2 · U · d1	[3]
  d1 + d2 = 1	[4]

• See Section General constraints for an alternative formulation via general constraints