EVOLUTIONARY OPTIMIZATION ALGORITHMSBy

Sumanth (EC8435)

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OVERVIEW

What is Optimization? How to Optimize? Deterministic Techniques Vs

Evolutionary Techniques Genetic Algorithm Particle Swarm Optimization Ant Colony Optimization Differential Evolution Conclusion ( Most efficient algorithm) References

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WHAT IS OPTIMIZATION ?

Effective selection of optimal solution. Maximum or Minimum. Cost function.

Applications Signal Processing. Communications. Computer Networks. Economics etc..

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HOW TO OPTIMIZE ??

Calculus tell us to differentiate and find optimum value.

To differentiate cost function must be known. Numerical methods can be used, if cost

function is not available. Newton’s forward and backward differences

can be used if sample data is known.

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MAJOR PROBLEM

As complexity of cost function increases.

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Rastrigin’s Function

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PROBLEMS WITH DETERMINISTIC TECHNIQUES

Struck at local minima or maxima. Computationally complex.

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local

global

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PROBABILISTIC OPTIMIZATION TECHNIQUES

Searches N dimensional search space randomly and lands on a approximate solution.

Doesn’t provide ideal optimum value but gives approximate value.

Evolutionary algorithms got their basic ideas from nature.

Examples Survival of the fittest. Motion of birds in search of food. Ants in search of food. Bees etc..

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DETERMINISTIC VS PROBABILISTIC TECHNIQUES

Results when Rastrigin's function is optimized in both methods.

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Compared value

Deterministic Technique

Genetic Algorithm Particle Swarm Optimization

Maximum 60.3002 76.5207 72.042

Minimum 3.1198 0.0095 1.3590

Time (in s) 74.84 0.1307 0.0564

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GENETIC ALGORITHM (GA)

Proposed by John Holland in 1975. Based on natural selection process Survival

of the fittest. Every possible solution is called

chromosome. Chromosomes are crossed over and mutated

to get better offsprings. Works iteratively. Each iteration is called as Generation. Best chromosomes are passed on to next

generations.

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GENETIC ALGORITHM (GA) CONTD.. 3

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Evaluation of Individuals

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GENETIC ALGORITHM (GA) CONTD..

Chromosome Binary coded - 1 0 1 0 0 1 1 1 0 1 Real encoded - 1.236 2.36 4.23

Crossover Single point Two point

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Chromosome 1

1 1 0 1 0 0 1 1 1 0 0

Chromosome 2

0 1 1 0 0 0 1 0 0 1 0

Offspring 1 1 1 0 1 0 0 1 0 0 1 0

Offspring 2 0 1 1 0 0 0 1 1 1 0 0

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GENETIC ALGORITHM (GA) CONTD..

Mutation

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0 5 10 15 20 2568

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No of iterations

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function

Convergence plot for Rastrigin’s Funciton

GA

chromosome

1 0 1 1 0 0 0 1 1 0

offspring 1 0 1 1 0 1 0 1 1 0

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PARTICLE SWARM OPTIMIZATION

Inspired by the swarm of birds in search of food.

Considers both self and group knowledge while searching.

Initial population are called as particles. Each particle has a position and velocity

randomly assigned. Lesser computational complexity than GA. Positions and velocities are updated in every

iteration. No crossover and mutation operators.

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PARTICLE SWARM OPTIMIZATION CONT.. 3

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PARTICLE SWARM OPTIMIZATION CONT..

pbest is particle’s best position in previous generations.

gbest is best position of all particles in a generation.

Velocity update rule

where c1,c2 are learning factors [0 4]

Position update rule

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])[][(**2])[][(**1][]1[ ixigbestrandcixipbestrandciViV

]1[][]1[ iVixix

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PARTICLE SWARM OPTIMIZATION CONT.. 3

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0 5 10 15 20 2570

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No of iterations

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function

Convergence plot for Rastrigin’s Funciton

PSO

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ANT COLONY OPTIMIZATION

Inspired by behavior of ants in search of food.

Majorly used for finding optimal paths in complex graphs.

Ants lays a trail of pheromone along their path.

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ANT COLONY OPTIMIZATION CONT..

Consider a Travelling Sales Man (TSP) problem.

N cities and M ants. Condition for transition from a city i to j :

Whether it is already visited or not. Distance between the cities (dij ) . Amount of artificial pheromone.

Path with maximum pheromone is the shortest of all paths.

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DIFFERENTIAL EVOLUTION (DE)

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Initial population candidate solutions within search space called Agents.

Optimizes a problem by iteratively improving the candidate solutions.

Similar to GA , DE uses recombination and mutation operators.

Phases of DE : Initialization Mutation Recombination Selection

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DIFFERENTIAL EVOLUTION (DE) CONT..

Initialization XiG = [ X1,I,G , X2,i,G , X3,i,G, …. XD,I,G ,] i= 1,2,3 …

N Mutation Vi,G+1 =Xr1,G + F * (Xr2,G - Xr3,G)

where F is differential weight € [0 2] Recombination

Selection Selected agents are passed on to next

generations.

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Gij

GijGij X

Vu

,,

1,,1,,

If randji ≤ CR or j = Irand

If randji > CR or j ≠ Irand

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CONCLUSION

Which Optimization algorithm is best? Which algorithm should be used for my

problem?

NO FREE LUNCH theorem for optimization. Every optimization algorithm is best for one

or other problem. On an average all algorithms efficiency is

almost equal. Simulate and find which algorithm best suites

your problem.

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REFERENCES Global Optimization Algorithms Theory and Application: Thomas Weise,

Version: 2009-06-26. Marcin Molga, Czesław Smutnicki Test functions for optimization needs :

kwietnia 2005. Yi-Tung Kao, Erwie Zahara. "A hybrid genetic algorithm and particle

swarm optimization for multimodal functions" Science Direct, Applied Soft Computing 8 (2008) 849–857.

Y. Shi, R. Eberhart."Empirical study of particle swarm optimization ".Proc. of Congress on Evolutionary Computation, 1999: 1945-1950.

Tutorial on Particle Swarm Optimization - http://www.swarmintelligence.org/tutorials.php

Hozefa M Botee, Eric Bonabeau: Evolving Ant Colony Optimization: Adv Computer Systems (1998) 1, 149_159.

An Introduction to Differential Evolution: Kelly Fleetwood.

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