D Nagesh Kumar, IISc Optimization Methods: M3L61

Linear Programming

Other Algorithms

D Nagesh Kumar, IISc Optimization Methods: M3L62

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 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: M3L63

Comparative discussion between new methods and Simplex method

Simplex AlgorithmKarmarkar’s Algorithm

Optimal solution point

Feasible Region

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.

D Nagesh Kumar, IISc Optimization Methods: M3L64

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

D Nagesh Kumar, IISc Optimization Methods: M3L65

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. 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: M3L66

Karmarkar’s projective scaling method

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: M3L67

Karmarkar’s projective scaling method

LP problems should be expressed in the following form:

:with

1

:subject to

Minimize T

0X

1X

0AX

XC

Z

where

nx

x

x

2

1

X

nc

c

c

2

1

C n 1111 1

mnmm

n

n

ccc

ccc

ccc

21

22221

11211

A 2nand

D Nagesh Kumar, IISc Optimization Methods: M3L68

Karmarkar’s projective scaling method

It is also assumed that is a feasible solution and

Two other variables are defined as: and .

n

n

n

1

1

1

0

X 0min Z

1

1

nnr

n

n

3

1

Karmarkar’s projective scaling method follows iterative steps to find the optimal solution

D Nagesh Kumar, IISc Optimization Methods: M3L69

Karmarkar’s projective scaling method

In general, kth iteration involves following computations

a) Compute

where

1

ADP

k

kDCC T

nk

k

k

k

X

X

X

D

000

000

0020

0001

If , ,any feasible solution becomes an optimal solution. Further iteration is not required. Otherwise, go to next step.

0C p

D Nagesh Kumar, IISc Optimization Methods: M3L610

Karmarkar’s projective scaling method

b)p

pnew r

C

CXY 0

c)newk

newkk Y1D

YDX 1

However, it can be shown that for k =0, k newnew

k new

D Y

Y1D Y

Thus, 1 newX Y

1T

kZ XCd)

Repeat the steps (a) through (d) by changing k as k+1e)

D Nagesh Kumar, IISc Optimization Methods: M3L611

Karmarkar’s projective scaling method: Example

Consider the LP problem:

0,,

1

02 :subject to

2 Minimize

321

321

321

32

xxx

xxx

xxx

xxZ

3nThus,

1

2

0

C 121 A

31

31

31

0

X

6

1

133

1

1

1

nnr

9

2

33

13

3

1

n

nand also,

D Nagesh Kumar, IISc Optimization Methods: M3L612

Karmarkar’s projective scaling method: Example

Iteration 0 (k=0):

3/100

03/10

003/1

0D

3/13/20

3/100

03/10

003/1

1200T

DCC

3/13/23/1

3/100

03/10

003/1

1210

AD

D Nagesh Kumar, IISc Optimization Methods: M3L613

Karmarkar’s projective scaling method: Example

Iteration 0 (k=0)…contd.:

111

3/13/23/10

1

ADP

30

03/2

13/1

13/2

13/1

111

3/13/23/1TPP

3/10

05.11TPP

D Nagesh Kumar, IISc Optimization Methods: M3L614

Karmarkar’s projective scaling method: Example

5.005.0

010

5.005.01TT PPPP

Iteration 0 (k=0)…contd.:

D Nagesh Kumar, IISc Optimization Methods: M3L615

Karmarkar’s projective scaling method: Example

Iteration 0 (k=0)…contd.:

3974.0

3333.0

2692.0

6/1

0

6/1

6

2

6

1

9

2

31

31

31

0p

pnew r

C

CXY

3974.0

3333.0

2692.0

1 newYX 2692.0

3974.0

3333.0

2692.0

1201T

XCZ

D Nagesh Kumar, IISc Optimization Methods: M3L616

Karmarkar’s projective scaling method: Example

Iteration 1 (k=1):

3974.000

03333.00

002692.0

1D

T1

0.2692 0 0

0 2 1 0 0.3333 0 0 0.6667 0.3974

0 0 0.3974

C C D

3974.06666.02692.0

3974.000

03333.00

002692.0

1211

AD

D Nagesh Kumar, IISc Optimization Methods: M3L617

Karmarkar’s projective scaling method: Example

Iteration 1 (k=1)…contd.:

1 0.2692 0.6667 0.3974

1 1 1

ADP

1

T

0.2692 10.2692 0.6667 0.3974 0.675 0

0.6667 11 1 1 0 3

0.3974 1

PP

1T T

0.441 0.067 0.492

0.067 0.992 0.059

0.492 0.059 0.567

P PP P

D Nagesh Kumar, IISc Optimization Methods: M3L618

Karmarkar’s projective scaling method: Example

Iteration 1 (k=1)…contd.:

0

1 30.151 0.26532 1

1 3 9 6 0.018 0.34140.2014

0.132 0.39281 3

pnew

p

r

CY X

C

D Nagesh Kumar, IISc Optimization Methods: M3L619

Karmarkar’s projective scaling method: Example

Iteration 1 (k=1)…contd.:

1

0.2692 0 0 0.2653 0.0714

0 0.3333 0 0.3414 0.1138

0 0 0.3974 0.3928 0.1561

new

D Y

1

0.0714

1 1 1 0.1138 0.3413

0.1561

new

1D Y

D Nagesh Kumar, IISc Optimization Methods: M3L620

Karmarkar’s projective scaling method: Example

Iteration 1 (k=1)…contd.:

12

1

0.0714 0.20921

0.1138 0.33340.3413

0.1561 0.4574

new

new

D YX

1D Y

T2

0.2092

0 2 1 0.3334 0.2094

0.4574

Z

C X

Two successive iterations are shown. Similar iterations can be followed to get the final solution upto some predefined tolerance level

D Nagesh Kumar, IISc Optimization Methods: M3L621

Thank You