1. Duality Dual problem Duality Theorem Complementary Slackness
Economic interpretation of dual variables

2. Primal Problem max +4 x1 +1 x2 +5 x3 +3 x4 s.t. +1 x1 -1 x2 -1 x3 +3≤ 1 +5 x1 +1 x2 +3 x3 +8 x4 ≤ 55 -1 x1 +2 x2 +3 x3 -5 x4 ≤ 3

3. Vector View max +4 x1 +1 x2 +5 x3 +3 x4 s.t. +1 -1 -1 +3 1 +5 x1 +1 x2+8 x4 ≤ 55 -1 +2 +3 -5 3

4. Linearly combine rows max +4 x1 +1 x2 +5 x3 +3 x4 s.t. y1 +1 -1 -1 +3+8 x4 ≤ 55 y3 -1 +2 +3 -5 3

5. Move y into vectors max +4 x1 +1 x2 +5 x3 +3 x4 s.t. +1 y1 -1 y1 -1 y1+8 y2 x4 ≤ 55 y2 -1 y3 +2 y3 +3 y3 -5 y3 3 y3

6. Column Constraints max +4 +1 +5 +3 ≥ ≥ ≥ ≥ s.t. +1 y1 -1 y1 -1 y1 +3+8 y2 ≤ 55 y2 -1 y3 +2 y3 +3 y3 -5 y3 3 y3

7. Dual Problem min +1 y1 +55 y2 +5 y3 s.t. +1 y1 +5 y2 -1 y3 ≥ 4 -1 y1+2 y3 ≥ 1 -1 y1 +3 y2 +3 y3 ≥ 5 +3 y1 +8 y2 -5 y3 ≥ 3

8. Observations
The objective value of any feasible primal solution is less than the objective value of any feasible dual solution
Duality Theorem
If both problems have an optimal solution, they are equal in value
Optimal dual solution can be read off of final dictionary
Dual solution serves as a certificate of optimality
Quick verification of optimality of primal solution

9. Example from Text Maximize 4x1 + x2 + 5x3 + 3x4
s.t x1 - x2 - x3 + 3x4 ≤ 1
5x1 + x2 + 3x3 + 8x4 ≤ 55
-x1 + 2x2 + 3x3 - 5x4 ≤ 3
Final Dictionary
x2 = x1 - 4x3 - 5x5 + 3x7
x4 = x1 - x3 - 2x5 - x7
x6 = x1 + 9x3 +21x5 +11x7
z = x1 - 2x3 -11x5 - 6x7

10. Reading optimal solution to dual problem
Final Dictionary
x2 = x1 - 4x3 - 5x5 + 3x7
x4 = x1 - x3 - 2x5 - x7
x6 = x1 + 9x3 +21x5 +11x7
z = x1 - 2x3 -11x5 - 0x6 -6x7
Dual objective: min y1 + 55y2 + 3y3
Linking slack variables with dual variables
x5 associated with y1 → y1 = 11
x6 associated with y2 → y2 = 0
x7 associated with y3 → y3 = 6

11. Linking back to original problem
z = x1 - 2x3 -11x5 - 0x6 -6x7
Dual objective: min y1 + 55y2 + 3y3
y1 = 11
y2 = 0
y3 = 6
Original Primal Problem
Maximize 4x1 + x2 + 5x3 + 3x4
s.t x1 - x2 - x3 + 3x4 ≤ 1
5x1 + x2 + 3x3 + 8x4 ≤ 55
-x1 + 2x2 + 3x3 - 5x4 ≤ 3

12. Complementary Slackness 1
max +4 x1 +1 x2 +5 x3 +3 x4
s.t. +1 y1 -1 y1 -1 y1 +3 y1 1 y1 +5 y2 x1 +1 y2 x2 +3 y2 x3 +8 y2 x4 ≤ 55 y2 -1 y3 +2 y3 +3 y3 -5 y3 3 y3
+1y1 + 5y2 – 1y3 = 4 OR x1 = 0

13. Complementary Slackness 2
max +4 x1 +1 x2 +5 x3 +3 x4
s.t. +1 y1 -1 y1 -1 y1 +3 y1 1 y1 +5 y2 x1 +1 y2 x2 +3 y2 x3 +8 y2 x4 ≤ 55 y2 -1 y3 +2 y3 +3 y3 -5 y3 3 y3
+1x1 - x2 – x3 +3x4 = 1 OR y1 = 0

14. Checking Optimality Without Certificate
Given candidate primal solution x
Write down equation in dual variables y
Use dual constraints at equality corresponding to components of x ≠ 0
Add in equation yj = 0 if primal constraint i is not tight
If solution is a nondegenerate basic solution, there is a unique solution y

15. Example from Text Maximize 4x1 + x2 + 5x3 + 3x4
s.t x1 - x2 - x3 + 3x4 ≤ 1
5x1 + x2 + 3x3 + 8x4 ≤ 55
-x1 + 2x2 + 3x3 - 5x4 ≤ 3
Candidate solution
x1 = 0, x2 = 14, x3 = 0, x4 = 5
System of constraints
1 = -y1 + y2 + 2y3
3 = 3y1 + 8y2 - 5y3
0 = y2

16. Economic interpretation of dual variables
Values of optimal dual variables (yi) give the marginal value of small increases or decreases of the given resource (bi)
Requires optimal basic solution to be a nondegenerate basic optimal solution
See worksheet for examples