1. Duality Theory
對偶理論

2. One of the most important discoveries in the early development of linear programming was the concept of duality.
Every linear programming problem is associated with another linear programming problem called the dual.
The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways.

3. Primal and Dual Problems
Primal Problem
Dual Problem
Max
s.t.
Min
s.t.
for
for
for
for
The dual problem uses exactly the same parameters as the primal problem, but in different location.

4. In matrix notation
Primal Problem
Dual Problem
Maximize
subject to
Minimize
subject to
Where and are row vectors but and are column vectors.

5. Example
Primal Problem
in Algebraic Form
Dual Problem
in Algebraic Form
Max
s.t.
Min
s.t.

6. Primal Problem
in Matrix Form
Dual Problem
in Matrix Form
Max
s.t.
Min
s.t.

7. Primal-dual table for linear programming
Primal Problem
Coefficient of:
Right
Side
Coefficient
of:
Objective Function
Coefficients for
(Minimize)
Dual Problem VI VI VI
Right
Side
Coefficients for
Objective Function
(Maximize)

8. Relationships between Primal and Dual Problems
One Problem Other Problem
Constraint Variable
Objective function Right sides
Minimization
Maximization
Variables
Constraints
Unrestricted
Constraints
Variables
Unrestricted

9. The feasible solutions for a dual problem are those that satisfy the condition of optimality for its primal problem.
A maximum value of Z in a primal problem equals the minimum value of W in the dual problem.

10. Rationale: Primal to Dual Reformulation
Lagrangian Function
Max cx
s.t. Ax b
x 0
L(X,Y) = cx - y(Ax - b)
= yb + (c - yA) x
= c-yA
Min yb
s.t. yA c
y 0

11. The following relation is always maintained
yAx yb (from Primal: Ax b)
yAx cx (from Dual : yA c)
From (1) and (2), we have (Weak Duality)
cx yAx yb
At optimality
cx* = y*Ax* = y*b
is always maintained (Strong Duality).
(1)
(2)
(3)
(4)

12. “Complementary slackness Conditions” are obtained from (4)
( c - y*A ) x* = 0
y*( b - Ax* ) = 0
xj* > y*aj = cj , y*aj > cj xj* = 0
yi* > aix* = bi , ai x* < bi yi* = 0
(5)
(6)

13. Any pair of primal and dual problems can be converted to each other.
The dual of a dual problem always is the primal problem.

14. Converted to Standard Form
Dual Problem
Min W = yb,
s.t yA c y
Max (-W) = -yb,
s.t yA -c y
Converted to
Standard Form
Its Dual Problem
Max Z = cx,
s.t Ax b x
Min (-Z) = -cx,
s.t Ax -b x

15. Min
s.t.
Min
s.t.

16. Max
s.t.
Max
s.t.

17. “Complementary Slackness Conditions”.
Application of
“Complementary Slackness Conditions”.
Example: Solving a problem with 2 functional constraints by graphical method.
Optimal solution：
x1=10
x2=0
x3=0