Linear Programming
Other Algorithms
1
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Introduction & Objectives
Few other methods, for solving LP problems, use
an entirely different algorithmic philosophy.
–
–
Khatchian’s ellipsoid method
Karmarkar’s projective scaling method
Objectives


2
To present a comparative discussion between new
methods and Simplex method
To discuss in detail about Karmarkar’s projective
scaling method
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Comparative discussion between new
methods and Simplex method
Simplex Algorithm
Karmarkar’s Algorithm
Optimal solution point
Feasible Region
3
D Nagesh Kumar, IISc
Khatchian’s
ellipsoid method
and Karmarkar’s
projective scaling
method seek the
optimum solution
to an LP problem
by moving through
the interior of the
feasible region.
Optimization Methods: M3L6
Comparative discussion between new
methods and Simplex method
1.
Both Khatchian’s ellipsoid method and
Karmarkar’s projective scaling method have been
shown to be polynomial time algorithms.
Time required for an LP problem of size n is at most anb ,
where a and b are two positive numbers.
2.
Simplex algorithm is an exponential time algorithm
in solving LP problems.
Time required for an LP problem of size n is at most c2n,
where c is a positive number
4
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Comparative discussion between new
methods and Simplex method
3.
For a large enough n (with positive a, b and c), c2n >
anb.
The polynomial time algorithms are computationally superior to
exponential algorithms for large LP problems.
4.
5
However, the rigorous computational effort of
Karmarkar’s projective scaling method, is not
economical for ‘not-so-large’ problems.
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling method
6

Also known as Karmarkar’s interior point LP algorithm

Starts with a trial solution and shoots it towards the
optimum solution

LP problems should be expressed in a particular form
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling method
LP problems should be expressed in the following form:
Minimize
Z  CT X
subject to :
AX  0
1X  1
with : X  0
where
 x1 
 c1 
 
c 
 x2 
2
X  C 


 
 
c n 
 x n 
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1  1 1  1 1n 
D Nagesh Kumar, IISc
 c11
c
A   21
 

c m1
c12
c 22

cm2
 c1n 
 c 2 n 
and n  2
  

 c mn 
Optimization Methods: M3L6
Karmarkar’s projective scaling method
1 n 
 
1 n 
It is also assumed that X 0    is a feasible solution and Z min  0
  
 
1 n 
n  1
1
Two other variables are defined as: r 
and  
.
3
n
nn  1
Karmarkar’s projective scaling method follows iterative steps to
find the optimal solution
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D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling method
In general, kth iteration involves following computations
a)
Compute


C p  I  P T PP T
where
 AD k 

P
 1 


C  CT Dk

1

P CT
0
0
0 
 X k 1


 0
X k 2  0
0 

Dk  
 0
0

0 


 0
0
0 X k n 
If , C p  0 , any feasible solution becomes an optimal solution.
Further iteration is not required. Otherwise, go to next step.
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D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling method
b) Ynew  X 0   r
c)
X k 1 
Cp
Cp
D k Ynew
1D k Ynew
However, it can be shown that for k =0,
10
Dk Ynew
 Ynew Thus, X1  Ynew
1Dk Ynew
d)
Z  C T X k 1
e)
Repeat the steps (a) through (d) by changing k as k+1
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Z  2 x 2  x3
Consider the LP problem: Minimize
subject to :
x1  2 x 2  x3  0
x1  x 2  x3  1
x1 , x 2 , x3  0
0
 
Thus, n  3 C   2  A  1  2 1
 
 1
1 3
 
1 3
X0   
  
 
1 3
1

and also, r 
11
nn  1

1
33  1
D Nagesh Kumar, IISc

1
6
n  1  3  1  2
3n
3 3
9
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 0 (k=0):
0 
1 / 3 0


D0   0 1/ 3 0 


 0
0 1 / 3
0 
1 / 3 0


T
C  C D 0  0 2  1  0 1 / 3 0   0 2 / 3  1 / 3


 0
0 1 / 3
12
0 
1 / 3 0


AD 0  1  2 1  0 1 / 3 0   1 / 3  2 / 3 1 / 3


 0
0 1 / 3
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 0 (k=0)…contd.:
 AD 0  1 / 3  2 / 3 1 / 3

P

 1   1
1
1 

 
 1 / 3 1
  2 / 3 0
1 / 3  2 / 3 1 / 3 
T
PP  
   2 / 3 1  

  0 3
 1
1
1  
 1 / 3 1
PP 
T 1
13
1.5 0 


 0 1 / 3
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 0 (k=0)…contd.:

P T PP T
 0.5 0 0.5


1
P 0 1 0 


0.5 0 0.5



C p  I  P T PP T
 1/ 6 


1
T
P C  0 


 1 / 6


C p  (1 / 6) 2  0  (1 / 6) 2 
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D Nagesh Kumar, IISc
2
6
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 0 (k=0)…contd.:
Ynew  X 0   r
X1  Ynew
15
0.2692


 0.3333


0.3974
Cp
Cp
1 3
  2 1  1 / 6  0.2692
1 3 9  6 
 

 
  0   0.3333
2
  

 

 
 1 / 6 0.3974
6
1 3
0.2692


T
Z  C X1  0 2  1 0.3333  0.2692


0.3974
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 1 (k=1):
0
0 
0.2692


D1   0
0.3333
0 


 0
0
0.3974
0
0 
 0.2692


T
C  C D1   0 2 1   0
0.3333
0   0 0.6667 0.3974 


 0
0
0.3974 
0
0 
0.2692
AD1  1  2 1  0
0.3333
0   0.2692  0.6666 0.3974
 0
0
0.3974
16
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 1 (k=1)…contd.:
 AD1  0.2692 0.6667 0.3974
P


 1   1
1
1 

 
 0.2692 1
 0.675 0 
 0.2692 0.6667 0.3974  
T
PP  
   0.6667 1  

  0
 1
1
1  
3 
 0.3974 1
0.492
 0.441 0.067



1
T
T
P  PP  P   0.067 0.992 0.059 


 0.492 0.059 0.567 
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D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 1 (k=1)…contd.:
C p  I  P T


 0.151 


1
T
T

PP
P C   0.018 



 0.132 

C p  (0.151)2   0.018  (0.132)2  0.2014
2
Ynew  X0   r
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Cp
Cp
1 3
  2 1  0.151   0.2653
1 3 9 

 

6
  
  0.018  0.3414
  0.2014 
 

 
 0.132   0.3928
1 3
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 1 (k=1)…contd.:
D1Ynew
0
0   0.2653 0.0714 
 0.2692

 
 

 0
0.3333
0   0.3414    0.1138 

 
 

 0
0
0.3974   0.3928   0.1561
1D1Ynew
19
 0.0714 


 1 1 1   0.1138   0.3413


 0.1561
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Karmarkar’s projective scaling
method: Example
Iteration 1 (k=1)…contd.:
X2 
D1Ynew
1D1Ynew
 0.0714  0.2092 

 

1

  0.1138   0.3334 
0.3413 
 

 0.1561 0.4574 
 0.2092 


T
Z  C X 2   0 2 1  0.3334   0.2094


 0.4574 
Two successive iterations are shown. Similar iterations can be followed
to get the final solution upto some predefined tolerance level
20
D Nagesh Kumar, IISc
Optimization Methods: M3L6
Thank You
21
D Nagesh Kumar, IISc
Optimization Methods: M3L6