Week 1 : Modelling tricks

Question 1

Either-or restrictions

Answer 1

a_1^Tx ≤ b1 ∨ a_2^Tx ≤ b2
Becomes:
0 ≤ x ≤ u
a_1^Tx − b1 ≤ M1(1 − z1)
a_2^Tx−b2 ≤ M2(1 − z2)
z1 + z2 = 1
zi ∈ B

with
Mi ≥ max{a_i^Tx−bi : 0≤ x ≤u}

Or:
a_1^Tx − b1 ≤ M1(1 − z)
a_2^Tx−b2 ≤ M2 z

Question 2

If-then restriction

Answer 2

0 ≤ x ≤ u
If a_1^T x > b1 then a_2^⊤x ≤ b2
is the same as 0 ≤ x ≤ u
¬a_1^⊤ x > b1 ∨ a_2^⊤x ≤ b