Date post: | 02-Jun-2018 |
Category: |
Documents |
Upload: | rookieanalytics |
View: | 228 times |
Download: | 0 times |
of 17
8/11/2019 4 Duality Theory
1/17
Duality Theory
8/11/2019 4 Duality Theory
2/17
One of the most important discoveries in the
early development of linear programming wasthe concept of duality.
Every linear programming problem is associated
with another linear programming problem calledthe dual.
The relationships between the dual problem and
the original problem (called the primal) prove
to be extremely useful in a variety of ways.
8/11/2019 4 Duality Theory
3/17
The dual problem uses exactly the same parameters
as the primal problem, but in different location.
Primal and Dual Problems
Primal Problem Dual Problem
Max
s.t.
Min
s.t.
n
j
jjxcZ1
,
m
i
iiybW1
,
n
j
ijij bxa1
, m
i
jiij cya1
,
for for.,,2,1 mi .,,2,1 nj
for .,,2,1 mi for .,,2,1 nj ,0jx ,0iy
8/11/2019 4 Duality Theory
4/17
In matrix notation
Primal Problem Dual Problem
Maximize
subject to
.0x .0y
Minimize
subject to
bAx cyA
,cxZ ,ybW
Where and are row
vectors but and are column vectors.
c myyyy ,,, 21
b x
8/11/2019 4 Duality Theory
5/17
Example
Max
s.t.
Min
s.t.
Primal Problemin Algebraic Form Dual Problemin Algebraic Form
,53 21 xxZ
,18124 321 yyyW
1823 21 xx
122 2x41x
0x,0x 21 522 32
yy
33 3 y1y
0y,0y,0y 321
8/11/2019 4 Duality Theory
6/17
Max
s.t.
Primal Problemin Matrix Form
Dual Problemin Matrix Form
Min
s.t.
,5,32
1
x
xZ
18
12
4
,
2
2
0
3
0
1
2
1
x
x
.0
0
2
1
x
x .0,0,0,, 321 yyy
5,3
2
2
0
3
0
1
,, 321
yyy
18
124
,, 321 yyyW
8/11/2019 4 Duality Theory
7/17
Primal-dual table for linear programming
Primal Problem
Coefficient of:RightSide
Righ
t
SideD
ualProblem
Coefficient
of
:
my
y
y
2
1
21
11
a
a
22
12
a
a
n
n
a
a
2
1
1x 2x nx
1c 2c ncVI VI VI
Coefficients forObjective Function
(Maximize)
1b
mna2ma1ma
2b
mb
Coefficients
for
Obj
ectiveFunction
(Minimize)
8/11/2019 4 Duality Theory
8/17
One Problem Other ProblemConstraint Variable
Objective function Right sides
i i
Relationships between Primal and Dual Problems
Minimization Maximization
Variables
Variables
Constraints
Constraints
0
0
0
0
Unrestricted
Unrestricted
8/11/2019 4 Duality Theory
9/17
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 dualproblem.
8/11/2019 4 Duality Theory
10/17
Rationale: Primal to Dual Reformulation
Max cx
s.t. Ax b
x 0L(X,Y) = cx - y(Ax - b)
=yb + (c - yA) x
Min yb
s.t. yA c
y 0
Lagrangian Function )],([ YXL
X
YXL
)],([=c-yA
8/11/2019 4 Duality Theory
11/17
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)
8/11/2019 4 Duality Theory
12/17
Complementary slackness Conditions are
obtained from (4)
( c - y*A ) x* = 0
y*( b - Ax*) = 0
xj* > 0 y*aj= cj , y*aj> cj xj* = 0
yi* > 0 aix* = bi , ai x*
8/11/2019 4 Duality Theory
13/17
Any pair of primal and dual problems can be
converted to each other.
The dual of a dual problem always is the primal
problem.
8/11/2019 4 Duality Theory
14/17
Min W = yb,s.t. yA c
y 0.
Dual Problem
Max (-W) = -yb,s.t. -yA -c
y 0.
Converted toStandard Form
Min (-Z) = -cx,s.t. -Ax -b
x 0.
Its Dual Problem
Max Z = cx,s.t. Ax b
x 0.
Converted toStandard Form
8/11/2019 4 Duality Theory
15/17
Min
s.t.
64.06.065.05.0
7.21.03.0
21
21
21
xxxx
xx
0,0 21 xx
21 5.04.0 xx
Min
s.t.
][y64.06.0
][y65.05.0
][y65.05.0
][y7.21.03.0
321
-
221
221
121
xx
xx
xx
xx
0,0 21 xx
21 5.04.0 xx
8/11/2019 4 Duality Theory
16/17
Max
s.t.
.0,0,0,0
5.04.0)(5.01.04.06.0)(5.03.0
6)(67.2
3221
3221
3221
3221
yyyy
yyyyyyyy
yyyy
Max
s.t.
.0,URS:,0
5.04.05.01.04.06.05.03.0
667.2
321
321
321
321
yyy
yyyyyy
yyy
8/11/2019 4 Duality Theory
17/17
Application of
Complementary Slackness Conditions.
Example: Solving a problem with 2 functional
constraints by graphical method.
0,,
104
3043..
372max
321
321
321
321
xxx
xxx
xxxts
xxxZ Optimal solution
x1=10
x2=0
x3=0