Minimum values

y = min{x₁, x₂} for two continuous variables x₁, x₂

• Must know lower and upper bounds

  L1 ≤ x1 ≤ U1	[1.1]
  L2 ≤ x2 ≤ U2	[1.2]

• Introduce binary variables d₁, d₂ to mean

  di	1 if xi is the minimum value;
  	0 otherwise

• MIP formulation:

  y ≤ x1	[2.1]
  y ≤ x2	[2.2]
  y ≥ x1 - (U1 - Lmin)(1 - d1)	[3.1]
  y ≥ x2 - (U2 - Lmin)(1 - d2)	[3.2]
  d1 + d2 = 1	[4]

• Generalization to y = min{x₁, x₂, ..., x_n}

  Li ≤ xi ≤ Ui	[1.i]
  y ≤ xi	[2.i]
  y ≥ xi - (Ui - Lmin)(1 - di)	[3.i]
  ∑i di = 1	[4]