The document describes the application of the Recursive Perturbation Approach for Multimodal Optimization (RePAMO) algorithm to various classical optimization problems. It summarizes the results of applying RePAMO to constrained Himmelblau, Griewank, Schwefel, Guilin Hills, Six Hump Camel, Rastrigin 3D, Ackley 4D, and Michalewicz 5D functions. For each function, it provides the number of minima and maxima found, the number of function evaluations, and the number of generations. Overall, the algorithm was able to successfully find the optima for all test functions.

1. Application of Recursive Perturbation Approach for
Multimodal Optimization (RePAMO) for classical
optimization problems
Presented by
Pritam Bhadra
Pranamesh Chakraborty
Indian Institute of Technology, Kanpur
11 May 2013

2. 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

3. Results
Constrained Himmelblau function
2 2 2 2
2 2
( , ) ( 11) ( 7)
subjected to
x 25
f x y x y x y
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
100
200
300
400
500
600
700
800
900
Figure 1: 3d plot of Himmelblau function

4. Constrained Himmelblau function
Figure 2: Connectivity of optima of Constrained Himmelblau finction

5. Constrained Himmelblau function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Contrained
Himmelblau
4 4 113250 6
The global minima is at (-3.775, -3.278) with f=0 and global
maxima at (0.26,-5).

6. Greiwank function
2
2
1 1
1
( ) cos 1
4000
X={x 600 600 (i=1,2)}
n
i
i
i i
i
x
f 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

7. Greiwank function
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.21144
0.81717
0.81717
0.81717
0.81717 0.81717
0.81717 0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
0.81717
1.4229
1.4229
1.4229
1.4229
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
Figure 4: Contour plot of Greiwank function

8. Greiwank function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Greiwank
function
1780 480 129,970 7
The global minima is at (0,0) with f=0 and global maxima at (600,600)

9. 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
-10
0
10
20
30
40
50
-50
-40
-30
-20
-10
0
10
20
30
40
50
-80
-60
-40
-20
0
20
40
60
80
Figure 5: 3d plot of Schwefel function

10. 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
-1.0357
-0.97093
-0.97093
-0.97093
-0.9062
-0.9062
-0.9062
-0.84147
-0.84147
-0.84147
-0.84147
-0.77674
-0.77674
-0.77674
-0.77674
-0.71201
-0.71201
-0.71201
-0.71201
-0.64729
-0.64729
-0.64729
-0.58256
-0.58256
-0.58256
-0.51783
-0.51783
-0.51783
-0.4531
-0.4531
-0.4531
-0.38837
-0.38837
-0.32364
-0.32364
-0.25891
-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 1
0
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

11. Schwefel function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Schwefel
function
19 14 198,978 8
The global minima is at (421,421) and global maxima at
(302.565,302.565)

12. Guilin Hills function
2
1
9
( ) 3 sin
110 1
2
X={x 0 1 (i=1,2)}
i
i
i i
i
i
i
x
f X c
x x
k
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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

13. Guilin Hills function
1.1411
1.1411
1.1411
1.5205
1.5205
1.5205
1.5205
1.5205
1.5205
1.5205
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
1.8999
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.2793
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
2.6587
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.0381
3.4176
3.4176
3.4176
3.4176
3.4176
3.4176
3.4176
3.4176
3.4176
3.797
3.797 3.797
3.797
3.797
4.1764
4.1764
4.5558
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
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

14. Guilin Hills function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Guilin Hills
function
15 25 278,978 6
The global minima is at (420.969,420.969)

15. Six Hump Camel function
6
2 4 2 41
1 1 1 2 2 2
1
2
( ) 4 2.1 4 4
3
: 2 2
1 1
x
f X x x x x x x
for x
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-2
-1
0
1
2
3
4
5
6
Figure 10: 3d plot of Six Hump Camel function

16. Six Hump Camel function
-0.83679
-0.67399
-0.67399
-0.51118
-0.51118
-0.34838
-0.34838
-0.34838
-0.18557
-0.18557
-0.18557
-0.022769
-0.022769
-0.022769
-0.022769
0.14004
0.14004
0.14004
0.30284
0.30284
0.30284
0.46565
0.46565
0.46565
0.62845
0.62845
0.62845
0.79126
0.79126
0.79126
0.95406
0.95406
0.95406
1.1169
1.1169
1.1169
1.2797
1.2797
1.2797
1.4425
1.4425
1.4425
1.6053
1.6053
1.6053
1.7681
1.7681
1.7681
1.9309
1.9309
1.9309
2.0937
2.0937
2.2565
2.2565
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 1
0
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

17. Six Hump Camel function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Six Hump
Camel
function
6 12 72,563 5

18. 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
function
8 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)

19. Higher dimensional problems
Ackley 4d function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Ackley 4d
function
333 378 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 1
0.2 cos(2 )
( ) 20 20 (i=1,2,....n)
X={x 2 2 (i=1,2,....n)}
n n
i i
i i
x x
n n
i
f X e e e
x

20. Higher dimensional problems
Michalewicz 5d function
Function
# Of
Minima
# Of
Maxima
# Of Function
Evaluation
# Of
Generation
Michalewicz
5d function
23,962 3,240 30,102,813 7
The global minima is at (2.203,1.571,1.285,1.114,1.72)
2
25
1
( ) sin sin
X={x 0 1 (i=1,5)}, m=10
m
i
i
i
i
ix
f X x
x

21. Function
# of
Minima
# of
Maxima
# of Function
Evaluation
# of Generation
Contrained
Himmelblau
4 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

22. 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.

23. THANK YOU