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Application of recursive perturbation approach for multimodal optimization
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.
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Application of recursive perturbation approach for multimodal optimization
- 1. Application of RecursivePerturbation 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-startalgorithm 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 22 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 Figure2: 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 0.817170.81717 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 0.10.2 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 3.7973.797 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 Camelfunction 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 Camelfunction -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 Camelfunction Function # Of Minima # Of Maxima # Of Function Evaluation # Of Generation Six Hump Camel function 6 12 72,563 5
- 18. Higher dimensional problems Rastrigin3d 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 Ackley4d 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 Michalewicz5d 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.
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