Application of Recursive Perturbation Approach for

Multimodal Optimization (RePAMO) for classical

optimization problems

Presented byPritam Bhadra

Pranamesh Chakraborty

Indian Institute of Technology, Kanpur

11 May 2013

Formulation of RePAMO

Multi-start algorithm dealing with variable population

A selected classical optimization method (in this case

Nelder Mead's Simplex Search Method) is recursively applied

to find all optima of a function.

The idea of climbing the hills and sliding down to the

nearby hills is applied.

Three basic operators:

1. Direction Set Generation and Perturbation

2. Optimization

3. Comparison

Results

Constrained Himmelblau function2 2 2 2

2 2

( , ) ( 11) ( 7)

subjected to

x 25

f x y x y x y

y

-5-4

-3-2

-10

12

34

5

-5-4

-3-2

-10

12

34

5

0

100

200

300

400

500

600

700

800

900

Figure 1: 3d plot of Himmelblau function

Constrained Himmelblau function

Figure 2: Connectivity of optima of Constrained Himmelblau finction

Constrained Himmelblau function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Contrained

Himmelblau4 4 113250 6

The global minima is at (-3.775, -3.278) with f=0 and global

maxima at (0.26,-5).

Greiwank function

22

1 1

1( ) cos 1

4000

X={x 600 600 (i=1,2)}

ni

i

i i

i

xf X x

i

x

-50

-40

-30

-20

-10

0

10

20

30

40

50

-50-40-30-20-1001020304050

-1

0

1

2

3

Figure 3: 3d plot of Griewank function

Greiwank function

0.2

11

44

0.21144

0.21144

0.21144

0.2

1144

0.2

11

44

0.2

11

44

0.21144

0.21144

0.2

11

44

0.21144

0.21144

0.8

17

17

0.8

17

17

0.8

17

17

0.817170.8

17

17

0.81717 0.81

717

0.81717

0.8

17

17

0.81717

0.8

17

17

0.81717

0.81717

0.81717

0.81717

0.81717

0.81717

0.81717

0.81717

0.81717

1.4

229

1.4

229

1.4

229

1.4

229

-60 -40 -20 0 20 40 60-60

-40

-20

0

20

40

60

Figure 4: Contour plot of Greiwank function

Greiwank function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Greiwank

function1780 480 129,970 7

The global minima is at (0,0) with f=0 and global maxima at (600,600)

Schwefel function

2

1

( ) sin

X={x 500 500 (i=1,2)}

i i

i

i

f X x x

x

-50-40

-30-20

-100

1020

3040

50

-50-40

-30-20

-100

1020

3040

50

-80

-60

-40

-20

0

20

40

60

80

Figure 5: 3d plot of Schwefel function

Schwefel function

-1.4888-1.424-1.3593

-1.2946-1.2298

-1.2298

-1.1651

-1.1651

-1.1004

-1.1004

-1.0357

-1.0357

-1.03

57

-0.97093

-0.97093

-0.97

093

-0.9062

-0.9062

-0.9062

-0.84147

-0.84147

-0.84147

-0.8

4147

-0.7

7674

-0.77674

-0.77674

-0.77674

-0.7

1201

-0.71201

-0.71201

-0.71201

-0.64

729

-0.64729

-0.64729

-0.58

256

-0.58256

-0.58256

-0.5

1783

-0.51783

-0.51783

-0.4

531

-0.4531

-0.4531

-0.38837

-0.38837

-0.32364

-0.32364

-0.25

891

-0.25891

-0.19419

-0.12946

-0.064729

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6: Contour plot of Schwefel function

Schwefel function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Schwefel

function19 14 198,978 8

The global minima is at (421,421) and global maxima at

(302.565,302.565)

Guilin Hills function

2

1

9( ) 3 sin

1101

2

X={x 0 1 (i=1,2)}

ii

i ii

i

i

xf X c

xx

k

x

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Figure 7: 3d plot of Guilin Hills function

Guilin Hills function

1.1

41

1

1.1411

1.1411

1.5

20

5

1.5205

1.5205

1.5

20

5

1.5

205

1.5205

1.5205

1.8999

1.8999

1.8

999

1.8999

1.8999

1.8999

1.8

99

9

1.8999

1.8999 1.8

99

91.8999 1.8

99

9

2.2793

2.2793

2.2793

2.2793

2.2793

2.2793

2.27932.2793

2.2

793

2.2793

2.2793

2.2793

2.2

79

3 2.2793

2.65872.6587

2.6587

2.65872.6587

2.6587

2.6587

2.6587

2.6587

2.6587

2.6587

2.6587

2.6

587

2.6

58

7

2.6587

3.0381

3.0

381

3.0

38

1

3.0381

3.0381

3.03813.0381

3.0381

3.0381

3.0381

3.0381

3.0381

3.4176 3.4

17

6 3.4

17

6 3.4176

3.4176

3.41763.4176

3.4176

3.4176

3.7

97

3.797 3.797

3.7

97

3.797

4.1

764

4.1

76

4

4.5

55

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 8: Contour plot of Guilin Hills function

Guilin Hills function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Guilin Hills

function15 25 278,978 6

The global minima is at (420.969,420.969)

Six Hump Camel function

62 4 2 411 1 1 2 2 2

1

2

( ) 4 2.1 4 43

: 2 2

1 1

xf X x x x x x x

for x

x

-2-1.5

-1-0.5

00.5

11.5

2

-1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

-2

-1

0

1

2

3

4

5

6

Figure 10: 3d plot of Six Hump Camel function

Six Hump Camel function

-0.8367

9

-0.67399

-0.67399

-0.5

1118

-0.51118

-0.34838

-0.3

4838

-0.34838

-0.18557

-0.1

8557

-0.18557

-0.022769

-0.0

22769

-0.0227

69

-0.022769

0.14004

0.1

40

04

0.14

004

0.30284

0.3

0284

0.3

02

84

0.46565

0.4

65

65

0.4

65

65

0.62845

0.6

28

45

0.6

28

45

0.79126

0.7

91

26

0.7

91

26

0.95406

0.9

54

06

0.9

54

06

1.1169

1.1

16

9

1.1

16

9

1.2797

1.2

79

7

1.2

79

7

1.4425

1.4

42

5

1.4

42

5

1.6053

1.6

05

3

1.6

05

3

1.7681

1.7

68

1

1.7

68

1

1.9309

1.9

30

9

1.9

30

9

2.0937

2.0

937

2.2565

2.2

56

5

2.4193

2.5821

2.7449

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11: Contour plot of Six Hump Camel function

Six Hump Camel function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Six Hump

Camel

function

6 12 72,563 5

Higher dimensional problems

Rastrigin 3d function

2

1

( ) (10 10cos(2 )

to

0.5 1.5

n

i i

i

i

f x x x

subjected

x

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Rastrigin 3d

function8 15 18,670 5

The global minima is at (0,0,0) with f=0 and global maxima at

(1.5,1.5,1.5)

Higher dimensional problems

Ackley 4d function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Ackley 4d

function333 378 12,624,183 9

The global minima is at (0,0,0,0) with f=0 and global maxima at

(2,1.6,1.6)

2

1 1

1 10.2 cos(2 )

( ) 20 20 (i=1,2,....n)

X={x 2 2 (i=1,2,....n)}

n n

i i

i i

x xn n

i

f X e e e

x

Higher dimensional problems

Michalewicz 5d function

Function# Of

Minima

# Of

Maxima

# Of Function

Evaluation

# Of

Generation

Michalewicz

5d function23,962 3,240 30,102,813 7

The global minima is at (2.203,1.571,1.285,1.114,1.72)

225

1

( ) sin sin

X={x 0 1 (i=1,5)}, m=10

m

ii

i

i

ixf X x

x

Function# of

Minima

# of

Maxima

# of Function

Evaluation# of Generation

Contrained

Himmelblau4 4 113250 6

Griewank 1780 480 1,29,970 7

Schwefel 14 19 1,98,978 8

Guilinhills 15 25 2,78,978 6

SixHumpCamel 6 12 72,563 5

Rastrigin 3-D 8 15 18,670 5

Ackley 4-D 333 378 1,26,24,183 9

Michalewicz 5-D 23962 3240 3,01,02,813 7

The algorithm worked successfully for all functions considered in this case.

Conclusions

Summary of results obtained

References

1. Bhaskar Dasgupta , Kotha Divya , Vivek Kumar Mehta & Kalyanmoy Deb

(2012):RePAMO: Recursive Perturbation Approach for Multimodal Optimization,

Engineering Optimization, DOI:10.1080/0305215X.2012.725050 aDepartment of

Mechanical engineering, IIT Kapur; bISRO Sattelite Centre, Bangalore.

THANK YOU