Engineering Optimisation Lecture
Dec 21, 2015
Lecture slides on Engineering Optimisation.
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(Rajesh Mishra)
Rajesh P Mishra lect 17 2
1. Given a LPP (called the primal problem), we shall
associate another LPP called the dual problem of the
original (primal) problem.
2. We shall see that the Optimal values of the primal and
dual are the same provided both have finite feasible
solutions.
3. This topic is further used to develop another method of
solving LPPs and is also used in the sensitivity (or
post-optimal) analysis.
Rajesh P Mishra lect 17 3
Dual Problem of an LPP
Definition of the dual problem
Given the primal problem (in standard form)
Maximize nnxcxcxcz ...2211
subject to 11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2 1 2
...
...
.
.
...
, ,..., 0, , ,..., 0
n n
n n
m m mn n m
n m
a x a x a x b
a x a x a x b
a x a x a x b
x x x b b b
Rajesh P Mishra lect 17 4
the dual problem is the LPP
Minimize mm ybybybw ...2211
subject to
11 1 21 2 1 1
12 1 22 2 2 2
1 1 2 2
1 2
...
...
.
.
...
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
Rajesh P Mishra lect 17 5
If the primal problem (in standard form) is
Minimize nnxcxcxcz ...2211
subject to 11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2 1 2
...
...
.
.
...
, ,..., 0, , ,..., 0
n n
n n
m m mn n m
n m
a x a x a x b
a x a x a x b
a x a x a x b
x x x b b b
Rajesh P Mishra lect 17 6
Then the dual problem is the LPP
Maximize mm ybybybw ...2211
subject to
11 1 21 2 1 1
12 1 22 2 2 2
1 1 2 2
1 2
...
...
.
.
...
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
Rajesh P Mishra lect 17 7
1. In the dual, there are as many (decision)
variables as there are constraints in the
primal.
We usually say yi is the dual variable
associated with the ith constraint of the
primal.
2. There are as many constraints in the dual
as there are variables in the primal.
We thus note the following:
Rajesh P Mishra lect 17 8
3. If the primal is maximization then the
dual is minimization and all constraints
are
If the primal is minimization then the dual
is maximization and all constraints are
4. In the primal, all variables are 0 while
in the dual all the variables are
unrestricted in sign.
Rajesh P Mishra lect 17 9
5. The objective function coefficients cj of
the primal are the RHS constants of the
dual constraints.
6. The RHS constants bi of the primal
constraints are the objective function
coefficients of the dual.
7. The coefficient matrix of the constraints
of the dual is the transpose of the
coefficient matrix of the constraints of
the primal. Rajesh P Mishra lect 17 10
Write the dual of the LPP
21 25 xxz
subject to
0,
532
2
21
21
21
xx
xx
xx
Maximize
Rajesh P Mishra lect 17 11
Thus the primal in the standard form is:
4321 0025 xxxxz
subject to
0,,,
532
2
4321
421
321
xxxx
xxx
xxx
Maximize
Rajesh P Mishra lect 17 12
Hence the dual is:
21 52 yyw
subject to
1 2
1 2
1
2
1 2
2 5
3 2
0
0
, unrestricted in sign
y y
y y
y
y
y y
0,0 21 yy
Minimize
Rajesh P Mishra lect 17 13
Write the dual of the LPP
21 36 xxz
subject to
0,,
543
236
321
321
321
xxx
xxx
xxx
Minimize
Rajesh P Mishra lect 17 14
Thus the primal in the standard form is:
54321 00036 xxxxxz
subject to
0,,,,
543
236
54321
5321
4321
xxxxx
xxxx
xxxx
Minimize
Rajesh P Mishra lect 17 15
Hence the dual is:
Maximize 21 52 yyw
subject to
1 2
1 2
1 2
1
2
1 2
6 3 6
3 4 3
0
0
0
, unrestricted in sign
y y
y y
y y
y
y
y y
0, 21 yy
Rajesh P Mishra lect 17 16
Write the dual of the LPP
21 xxz subject to
1 2
1 2
1 2
2 5
3 6
, unrestricted in sign
x x
x x
x x
Maximize
Rajesh P Mishra lect 17 17
Maximize 2211 xxxxz
subject to
0,,,
633
522
2211
2211
2211
xxxx
xxxx
xxxx
Thus the primal in the standard form is:
Rajesh P Mishra lect 17 18
Minimize 21 65 yyw
subject to
1 2
1 2
1 2
1 2
1 2
2 3 1
2 3 1
1
1
, unrestricted in sign
y y
y y
y y
y y
y y
132 21 yy
121 yy
Hence the dual is:
Rajesh P Mishra lect 17 19
From the above examples we get the
following SOB rules for writing the dual:
Label Maximization Minimization
Constraints Variables
Sensible form 0
Odd = form unrestricted Bizarre form 0
Variables Constraints
Sensible 0 form Odd unrestricted = form Bizarre 0 form
Rajesh P Mishra lect 17 20
Theorem: The dual of the dual is the primal
(original problem).
Proof. Consider the primal problem (in standard
form) Maximize nnxcxcxcz ...2211
subject to 11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2
...
...
.
.
...
, ,..., 0
n n
n n
m m mn n m
n
a x a x a x b
a x a x a x b
a x a x a x b
x x x
Rajesh P Mishra lect 17 21
The dual problem is the LPP
Minimize mm ybybybw ...2211
subject to 11 1 21 2 1 1
12 1 22 2 2 2
1 1 2 2
1 2
...
...
.
.
...
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
Rajesh P Mishra lect 17 22
Case (i): All cj 0. Then the dual problem in
the standard form is the LPP
Minimize n
mmmm
ttt
ybybybybw
0...00
...
21
1111
subject to
0,...,,,,,...,,
...
.
.
...
...
2111
1111
2222112112
1111111111
nmm
nnmmnmmnnn
mmmm
mmmm
tttyyyy
ctyayayaya
ctyayayaya
ctyayayaya
Rajesh P Mishra lect 17 23
Hence its dual is the LPP
Maximize nnxcxcxcz ...2211
subject to 11 1 12 2 1 1
11 1 12 2 1 1
1 1 2 2
1 1 2 2
1 2
1 2
...
...
.
...
...
0, 0,..., 0
, ,..., unrestricted in sign
n n
n n
m m mn n m
m m mn n m
n
n
a x a x a x b
a x a x a x b
a x a x a x b
a x a x a x b
x x x
x x x
Rajesh P Mishra lect 17 24
Which is nothing but the LPP
Maximize nnxcxcxcz ...2211
subject to
0,...,,
...
.
.
...
...
21
2211
22222121
11212111
n
mnmnmm
nn
nn
xxx
bxaxaxa
bxaxaxa
bxaxaxa
This is the primal problem. Rajesh P Mishra lect 17 25
Case (ii): All cj 0 except c1. Then the dual
problem in the standard form is the LPP
Minimize n
mmmm
ttt
ybybybybw
0...00
...
21
1111
Subj-
ect
to
0,...,,,,,...,,
...
.
.
...
...
2111
1111
2222112112
1111111111
nmm
nnmmnmmnnn
mmmm
mmmm
tttyyyy
ctyayayaya
ctyayayaya
ctyayayaya
Rajesh P Mishra lect 17 26
Hence its dual is the LPP
Maximize nnxcxcvcz ...2211
subject to 11 1 12 2 1 1
11 1 12 2 1 1
1 1 2 2
1 1 2 2
1 2
1 2
...
...
.
...
...
0, 0,..., 0
, ,..., unrestricted in sign
n n
n n
m m mn n m
m m mn n m
n
n
a v a x a x b
a v a x a x b
a v a x a x b
a v a x a x b
v x x
v x x
Rajesh P Mishra lect 17 27
Which is nothing but the LPP
Maximize nnxcxcvcz ...2211
subject to
0,...,,0
...
.
.
...
...
21
2211
22222121
11212111
n
mnmnmm
nn
nn
xxv
bxaxava
bxaxava
bxaxava
Putting 11 vx
Rajesh P Mishra lect 17 28
This is nothing but the LPP
Maximize nnxcxcxcz ...2211
subject to 11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2
...
...
.
.
...
, ,..., 0
n n
n n
m m mn n m
n
a x a x a x b
a x a x a x b
a x a x a x b
x x x
This is the primal problem. Cases where
other cj are 0 are similarly treated. Rajesh P Mishra lect 17 29
Duality theorems
Finding the dual optimal solution from the
primal optimal tableau
In this lecture we shall present the primal
and dual problems in matrix form and
prove certain results on the feasible and
optimal solutions of the primal and dual
problems.
Dual problem in Matrix form
Suppose the primal in (Matrix and ) standard
form is
0b0X
b,XA
,
subject to
Xcz
Dual problem in Matrix form
Maximize
where
nccc ..21C
nx
x
x
.
.
2
1
X, ,
mnmm
n
n
aaa
aaa
aaa
..
.
.
..
..
21
22221
11211
A
1
2
.
.
m
b
b
b
b
Letting myyy ..21Y
the dual LPP in matrix form becomes
Minimize bYw
subject to
Y A c,
Y unrestricted in sign
If the primal LPP is a minimization problem
the dual LPP in matrix form becomes
Maximize bYw
subject to
Y A c,
Y unrestricted in sign
Minimize Xcz
0b0X
b,XA
,
subject to
Rajesh P Mishra lect 17 36
Optimal Dual Solution • The primal and dual solutions are so closely
related that the optimal solution of either problem directly yields (with little additional computation) the optimal solution to the other.
• Thus, in an LP model in which the number of variables is considerably smaller than the number of constraints, computational savings may be realized by solving the dual, from which the primal solution is determined automatically
Rajesh P Mishra lect 17 37
Rajesh P Mishra lect 17 38
Rajesh P Mishra lect 17 39
Rajesh P Mishra lect 17 40
Rajesh P Mishra lect 17 41
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