Question

For each of the three sets below, find a missing valid inequality and verify graphically that its addition to the formulation gives conv(X).

(i) X = {x ∈ {0,1}²: 3x₁+x₂+x₃+2x₄ ≤ 6}

Answer 1

To find a missing valid inequality for the given set X = {x ∈ {0,1}²: 3x₁+x₂+x₃+2x₄ ≤ 6}, complete the square in x² and solve the resulting inequality. Verify the missing inequality graphically by plotting it with the original condition and checking for an intersection.

Step-by-step explanation:

To find a missing valid inequality for the given set X = {x ∈ {0,1}²: 3x₁+x₂+x₃+2x₄ ≤ 6}, we can start by completing the square in x². This will simplify the condition to 2(x² - 1)² ≤ 4. We can solve this inequality to obtain the missing valid inequality 2(x² - 1)² ≤ 4.

Graphically, we can plot the graph of the original condition and the missing valid inequality to see if they intersect. If they do, it means the missing inequality is valid. If they don't intersect, it means the missing inequality is not valid.