Department Of Industrial Engineering Duality And Sensitivity Analysis presented by: Taha Ben Omar Supervisor: Prof. Dr. Sahand Daneshvar

Introduction for every program we solve, there is another associated linear program which we happen to be simultaneously solving. The new linear program satisfies some very important properties. It may be used to obtain the solution to the original program. Its variables provides extremely useful information about the optimal solution to the original linear program.

Formulation of the Dual problem Canonical form of duality P: minimize cx Subject to Ax ≥ b X ≥ 0 D: Maximize wb Subject to wA ≤ c W ≥ 0

Example P: Minimize 6x 1 + 8x 2 Subject to 3x 1 + x 2 ≥ 4 5x 1 + 2x 2 ≥ 7 x 1, x 2 ≥ 0 D: Maximize 4w 1 + 7w 2 Subject to 3w 1 + 5w 2 ≤ 6 W 1 + 2w 2 ≤ 8 W 1, w 2 ≥ 0

Standard form of duality P: Minimize cx Subject to Ax = b X ≥ 0 D: Maximize wb Subject to wA ≤ c W unrestricted

Example P: Minimize 6x 1 + 8x 2 Subject to 3x 1 + x 2 – x 3 = 4 5x 1 +2x 2 - x 4 = 7 x 1, x 2, x 3, x 4 ≥ 0 D: Maximize 4w 1 +7w 2 Subject to 3w 1 + 5w 2 ≤ 6 w 1 + 2w 2 ≤ 8 -w 1 ≤ 0 - w 2 ≤ 0 w 1, w 2 unrestricted

Given one of the definitions canonical or standard, it is easy to demonstrate that the other definition is valid. For example suppose that we accept the standard form as a definition and wish to demonstrate that the canonical form is correct. Bu adding slack variables to the canonical form of a linear program, we may apply the standard form of duality to obtain the dual problem.

P: Max cx D: Max wb Subject to Subject to Ax –Ix = b wA ≤ c x x ≥ 0 w unrestricted since -wI ≤ 0 is the same w ≥ 0

Dual of the Dual Since the dual linear program is itself a linear program, we may wonder what its dual might be. Consider the dual in canonical form : Maximize wb Subject to wA ≤ c W ≤ 0 We may rewrite this problem in a different form : Minimize (-b t )w t Subject to (-A t )w t ≥ (-c t ) W t ≥ 0

The dual liner program for this linear program is given by ( letting x play the role of the row vector of dual variables) : - Maximize x t (-c t ) Subject to x t (-A t ) ≤ (-b t ) X t ≥ 0 But this is the same as: Minimize cx Subject to Ax ≥ b X ≥ 0 Which is precisely the primal problem. Thus we have the following lemma which is known as the involuntary property of Duality.

Lemma The dual of the dual is the primal. This lemma indicated that the definitions may be applied in reverse. The terms " primal" and "dual" are relative to the frame of reference we choose.

Mixed forms of Duality Consider the following linear program. P: Minimize c 1 x 1 + c 2 x 2 + c 3 x 3 Subject to A 11 x 1 + A 12 x 2 + A 13 x 3 ≤ b 1 A 21 x 1 + A 22 x 2 + A 23 x 3 ≤ b 2 A 31 x 1 + A 32 x 2 + A 3 x 33 = b 3 X 1 ≥ 0, x 2 ≤ 0, x 3 unrestricted

Converting this problem to conical form by multiplying the second set of inequalities by -1, write the equality constraint set equivalently as two inequalities, and substituting x 2 = -x' 2, x 3 = x' 3 – x 3 '' Minimize c 1 x 1 – c 2 x 2 + c 3 x 3 –c 3 x 3 Subject to A 11 x 1 – A 12 x 2 ' + A 13 x 3 ' – A 13 x 3 '' ≥ b 1 -A 21 x 1 + A 22 x 2 ' – A 23 x 3 ' + A 23 x 3 '' ≥ -b 2 A 31 x 1 – A 32 x 2 ' + A 33 x 3 ' – A 33 x 3 '' ≥ b 3 A 31 x 1 + A 32 x 2 ' - A 33 x 3 ' + A 33 x 3 '' ≥ -b 3 - X 1 >= 0, X' 2 >= 0, X' 3 ≥ 0, X 3 '' ≥ 0

Denoting the dual variable associated with the four constraints sets as w 1, w 2 '‘, w 3 '‘ and w 3 '‘respectively, we obtain the dual to this problem as follows. Minimize w 1 b 1 –w' 2 b 2 + w' 3 b 3 – w'' 3 b 3 Subject to w 1 A 11 – w' 21 A 1 + w' 31 A 1 – w'' 31 A 1 ≤ c - w 1 A 12 + w' 2 A 22 – w' 3 A 22 + w'' 3 A 32 ≤ c 2 w 1 A 13 – w' 2 A 23 + w' 3 A 33 – w'' 3 A 33 ≤ c 3 w 1 A 13 + w' 2 A 23 - w' 3 A 33 + w'' 3 A 33 ≤ c 3 - w' 1 >= 0, w' 2 >= 0, w' 3 >= 0, w'' 3 ≥ 0

MINIMIZATION PROBLEM ≤ 0 ≥ 0 = MAXIMIZATION PROBLEM ≥ 0 ≤ 0 Unrestricted ≥ 0 ≤ 0 Unrestricted ≤ 0 ≥ 0 =

Finally, using w 2 = -w 2 ' and w 3 = w 3 ' – w 3 '', the forgoing problem may be equivalently started as follows : D: Maximize w 1 b 1 +w 2 b 2 + w 3 b 3 Subject to w 1 A 11 + w 2 A 21 + w 3 A 31 ≤ c w 1 A 12 + w 2 A 22 + w 3 A 32 ≥ c w 1 A 13 + w 2 A 23 + w 3 A 33 = c w ≥ 0, w 2 ≤ 0, w 3 unrestricted

Example Consider the following linear program Maximize 8x 1 + 3x 2 + 2x 3 Subject to x 1 – 6x 2 + x 3 ≥ 2 5x 1 +7x 2 -2x 3 = -4 x 1 ≤ 0, x 2 ≥ 0, x 3 unrestricted

Applying the results of the table, we can immediately write down the dual. Minimize 2w 1 – 4w 2 Subject to w 1 +5w 2 <= 8 -6w 1 + 7w 2 ≥ 3 w 1 – 2w 2 = -2 w 1 <= 0, w 2 unrestricted