State and Prove Duality Theorem of linear programming and use it to prove the theorem of Complementary Slackness.

 The duality theorem states that:

• if the primal problem has an optimal solution, then so has the dual, and zP = zD; 1
• if the primal problem is unbounded, then the dual is infeasible;
• if the primal problem is infeasible, then the dual is either infeasible or unbounded.
Consider the following primal and dual problems :

Maximize zP = c T x     Minimize zD = b T y

subject to x ≥ 0, s ≥ 0    subject to y ≥ 0, t ≥ 0

Ax + s = b        AT y − t = c

For optimal solutions, complementary slackness states that yisi = 0 (i = 1, . . . , m), tjxj = 0 (j = 1, . . . , n).

 For feasible solutions of the primal and the dual, we have cj = 1, . . . , n

zP = c T x = (y TA − t T )x = y T (b − s) − t T x = zD − y T s − t T

For an optimal solution of the primal and dual, zP = zD
 so ,

y T s + t T x = 0

Since variables are non-negative this implies that :

yisi = 0 i = 1, . . . , m

 tjxj = 0 j = 1, . . . , n

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