A Hybrid Sequential Niche Genetic Algorithm for Multimodal Objective Functions

1. Midwest Regional Conference,2007 A Hybrid Sequential Niche Genetic Algorithm for Multimodal Objective Functions Jeonghwa Moon, Libin Zhang and Andreas A. Linninger Date :08/24/07 University of Illinois at Chicago, Chicago, IL Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. Motivation • Importance of global optimization in multimodal situation • Investigating and finding all physically meaningful solutions is important and challenging one in engineering and science field. • Multi-modal objective functions are common • Practical real problems have meaningful local minima • Two Kinds of optimization methods • We suggest “Hybrid algorithm for finding all solutions in multimodal function”. • Genetic algorithm + deterministic local optimizer • Based on sequential niche technique (Beasley 93)

3. Hybrid Genetic Algorithm • Why hybrid? • Accuracy of solution • Produce fast convergence • Two types of hybrid genetic algorithm Close coupling method Two stage search technique START START Initial population Initial population Repair Local optimizer Fitness evaluation Local optimizer Fitness evaluation Yes exact Solution Convergent? Yes Convergent? END Natural Selection END Natural Selection Mating Mating Mutation Mutation e.g: Li : GA+Gradient Sabatini: GA + Newton-Rhapson Sieary :GA + Simplex e.g: Hill-climbing operator (HCO)

4. Finding all solutions[#5] • The key point of finding all solutions in stochastic method is how to maintain population diversity. • Traditional GA converges to only primary solution • To do this, Niche formation methods were developed • Crowding (De Jong,75) • Deterministic crowding (Mahfoud, 95) • Fitness sharing (Goldberg,89) • Sequential niche technique (Beasley,93) • Dynamic niche method (Miller, 95)

5. Niches Niche Method • The meaning of niche • Different subspace that can support different types of life • In search space, each peak is Niche • The number of individuals supported by niche • Proportional to Niche capacity • Niche capacity is determined by peak fitness • This concept implementspopulation diversity

6. Fixed niche radius • The problem of fixed niche radius • Too small : individuals near the solution is recognized as another solution (Smith,92) • Too large : may lose some of solutions(Smith 92,Goldberg,92) • The value of niche radius(Deb,89)[#1] • Assumption : p is known, solutions are equally spread No of dimension No of solution

7. Some methods of tackling fixed radius problem • Hill-valley function (Ursem, 99) • Check whether two individuals are in one hill • Niche concept is not used • Methods using Variable Niche radius • Dynamic Niche Clustering (Gan, 00) • Covariance matrix Adaptation (Shir, 06) • But these cannot be combined with hybrid technique • Number of solution is needed, a-priori

8. Hybrid Genetic algorithm for multimodal optimization problem • Two stage optimizer • To guess initial point, stochastic method is used. • To get an exact solution, deterministic method is used. • It finds all solutions including local minima (maxima) sequentially • Hybrid version of sequential niche technique • Variable niche radius is used • Fitness sharing concept is used to detect ‘initial guess’ • With this initial guess, local deterministic optimizer is activated and finds an exact solution • No a priori- guess of number of solutions are needed • Small population size, but long iterations

9. Whole flowchart of algorithm START Generation of initial population Fitness evaluation Local optimizer Find exact Solution Initial guess point? Yes No Duplicated? Maximum iteration or terminated? No Yes END Update Radius Add solution Set Radius Natural Selection Mating Initialize population Mutation

10. Sequential Genetic Algorithm (Beasley 93) • It finds solutions sequentially • Derating functions Average of cost S2 found! S1 found! S3 found! No of iteration G :derating function F: raw fitness M: modified fitness a: power factor n: number of solution found

11. Condition for switching two methods • Condition for convergence (finding the region of optimum) • Select best individual Ib in current iteration • If the condition is satisfied, it switches to local optimizer and finds an exact solution m :niche count, ε: convergence factor, 0< ε <1 d: distance between Ij and Ib P: population size Sharing function

12. S2 P2 P1 S1 N1 Fixed Niche radius with HGA-trapping P1->S1 N1-is set up P2->S1 trapping P2->S1 P2->S1 … S2 is never found! Niche radius : r (fixed)

13. r2 r1 a Variable Niche radius • Radius is determined by • Distance between solution and initial guess • Radius is updated by r1 a P2 P1

14. P3 S2 P2 P1 S1 N1 N2 Variable Niche radius - example P1->S1 N1-is set up P2->S1 Area updating N1->N2 P3->S2 All minima are found! Niche radius : d(s,p)+a

15. Testing • Test functions • Deb’s functions • Himmelblau’s function • Grienwank’s function • Ackley’s function • Sequential GA + Steepest descent method is used

16. Maxima point Deb’s functions Pop size :20 Maximum iter. :100 • Equal Maxima • Decreasing Maxima • Uneven Maxima • Uneven Decreasing Maxima Initial guess Deb 1989, Beasley 1993

17. Himmelblau’s function individuals Initial guess solution Maxima (3.58,-1.86), (3.0,2.0), (-2.815,3.125),(-3.78,-3.28) Success rate of Beasley’s method :76% Our algorithm is 100%

18. Grienwank’s function Numerous global minimas! n=2

19. Ackley’s function 1 global minima and many unequal minimas! n=1 n=2 Result of simulation

20. Conclusion • New hybrid sequential genetic algorithms for multimodal optimization is suggested. • Derating method is used to give penalty • Fitness sharing is used to check condition for switching • Variable niche radius is used • Every solution has different size of niche radius • Niche area is updated when duplicate solution found. • small population size but long iteration required. • Several tests show it works very well

21. Future work • Suitable termination condition has to be researched • Different shape of Niche area • proper local optimizer • Stochastic (global) + deterministic (local) • Stochastic (global) + Stochastic (local) • Stochastic (global) + simplex (local) r Ellipse is more proper!

22. Reference • David Beasley David R Bull Ralph R Martin , Sequential Niche Technique for Multimodal Function Optimization , Evolutionary Computation 1(2), pp101-125 • David E Goldberg Jon Richardson (1987) Genetic algorithms with sharing for multimodal function optimization, In proceedings of the Second International Conference on genetic algorithms (pp 41-49) Lawrence Erlbaum Associates • Prasanna V Parthasarathy , David E Goldberg and Scott A Burns Tackling multimodal problems in hybrid genetic algorithms , Illegal Report No 20011012 March 2001 • Rachid Chelouah, Patrick Siarry, Genetic and Nelder-Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions , European Journal of Operational Research 148(2003) 335-348 • Brad L Miller, Michael J Shaw, Genetic algorithms with Dynamic Niche sharing for Multimodal function optimization , IEEE 1996 • Ofer M. Shir and Thomas Bäck. Niche Radius Adaptation in the CMA-ES Niching Algorithm - Parallel Problem Solving from Nature, PPSN IX, Reykjavik, Iceland; LNCS 4193

23. Questions?