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

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

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

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

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

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

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

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

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 = 14 - 2x1 - 4x3 - 5x5 + 3x7 x4 = 5 - x1 - x3 - 2x5 - x7 x6 = 1 + 5x1 + 9x3 +21x5 +11x7 z = 29 - x1 - 2x3 -11x5 - 6x7

Reading optimal solution to dual problem Final Dictionary x2 = 14 - 2x1 - 4x3 - 5x5 + 3x7 x4 = 5 - x1 - x3 - 2x5 - x7 x6 = 1 + 5x1 + 9x3 +21x5 +11x7 z = 29 - 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

Linking back to original problem z = 29 - 1x1 - 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

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

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

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

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

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