Genetic AlgorithmsGenetic Algorithms

Genetic Algorithms – What are they? And how they are inspired from evolution.

Operators and Definitions in Genetic Algorithms paradigm.

-chromosomes

-crossover, mutation and selection

-population, fitness and elitism Applications of Genetic Algorithms. Real Parameter Genetic Algorithms. Parent Centric Recombination Operator & G3 model.

Genetic algorithms are search algorithms based on the mechanics of natural selection and natural genetics (Goldberg 1989).

GA s exploit the idea of the survival of the fittest and an interbreeding population to create a novel and innovative search strategy.

In nature weaker members of a species tend to die away, leaving the stronger and fitter to mate, create offspring which evolve themselves into newer species and ensure the continuing survival of the fittest.

GA s are a form of randomized search, in that the way in which solutions are chosen and combined is a stochastic process rather than traditional deterministic problem solving techniques.

Operators and DefinitionsOperators and Definitions

Chromosome -- A possible solution to a problem represented traditionally as a binary string ( e.g. 11000110)

Crossover-- When two individuals mate, both parents pass their chromosomes onto their offspring. The two chromosomes come together and swap genetic material.

In binary GA s crossover is performed by swapping a part of binary strings between two solutions at a randomly chosen cross-site with some probability. It is a binary operator.

Operators and DefinitionsOperators and Definitions

Mutation – Conversion of proteins from one to another.

In Binary GA s mutation is performed by converting some random bit of a binary string into its complementary bit (ie a 1 to a 0 or vice versa) with some probability. It is a unary operator.

Mutation will help prevent the population from stagnating. It adds “fresh blood” to a population of solutions to a problem. It adds diversity.

Operators and DefinitionsOperators and Definitions

Fitness – The measure of goodness of a solution. (e.g. the function value of a solution in an optimization problem).

Selection – The Darwinian selection mechanism to eliminate bad solutions in a population.

Population – A pool of solutions represented as binary strings (chromosomes) that undergo

-- selection (based on their fitness) -- crossover (mutually among themselves randomly)

-- mutation (randomly) Note – crossover and mutation destroy old solutions Elitism – Some elite (good) solutions are carried onto

the next generation without being destroyed. It is considered to be a good strategy.

Roulette Wheel SelectionRoulette Wheel Selection

Solutions are selected on the basis of the percentage that their fitness contributes to the cumulative fitness of the whole population

GA flowchartGA flowchart

GA dynamicsGA dynamics

Applications of GA sApplications of GA s GA s are especially useful when -- The search space is large, complex or poorly understood. -- Domain knowledge is scarce or expert knowledge is difficult to encode to

narrow the search space. -- No mathematical analysis is available. -- Traditional search methods fail.Typically over the years GA s have been successfully applied to Function and Structural Optimization Database Query Optimization Determination of Protein Structures Scheduling Problems Construction and Training of Neural Networks Engineering Design Multi-criteria Decision Making (one of the hottest fields for GA s) Music Composition

Real Parameter Genetic AlgorithmsReal Parameter Genetic Algorithms

A number of real world applications involve object variables which are real valued

In the past few years researchers have tried to simulate the principles of crossover, mutation and selection in real-valued space directly.

A number of Real parameter crossover and mutation operators have been proposed:

-- Simulated Binary Crossover (SBX) (Deb 1995)

-- Evolution Strategies (ES) (Rechenberg & Schwefel)

-- Unimodal Normal Distribution Crossover (UNDX) (Kita 1997)

-- Simplex Crossover (SPX) (Yamamura 1997)

-- Parent Centric Crossover (PCX) (Deb 2001)

Real Coded CrossoversReal Coded Crossovers

UNDX -- It is a generic multi-parent crossover operator which emphasizes offspring closer to the geometric centroid of the parents.

SPX – It is a generic multi-parent crossover operator which uses a uniform probability to generate offspring inside a simplex determined by the parents.

PCX – It is a generic multi-parent crossover

operator which emphasizes offspring closer to the parents.

UNDX

SPX

PCX

Some Test Functions for Real GA sSome Test Functions for Real GA s

Schwefel’s Function

Ellipsoidal Function

Gen. Rosenbrock’s Function

n

i

j

i

j

x1

2

)(1

n

i

ixi1

*2

1

1

22 )( )1()( 12*100

n

ixxx iii

Generalized Generation Gap Model (G3 model)Generalized Generation Gap Model (G3 model)

The breeding scheme of Binary GA s is modified as follows From Population P, select the best parent and (M-1) other

parents randomly. Generate N offspring from the chosen M parents using

some recombination scheme. (UNDX, PCX etc). Choose two parents (p1 and p2) at random from the

population P. From the combined sub-population of p1, p2 and N

created offspring, choose the best two solutions and replace p1 and p2 with these two solutions.

Another modification is to replace only one parent in the population with best offspring. (Modified G3 model)

1

ResultsResults

Graphically presented are the number of function evaluations required to find a solution of fitness of 1e-20 for Felp as a function of the pool size (N here lambda).

A parametric study to find the optimal population size.

ResultsResults

Graphically presented are the number of function evaluations required to find a solution of fitness of 1e-20 for Fsch as a function of the pool size (N here lambda).

A parametric study to find the optimal population size.

ResultsResults

Graphically presented are the number of function evaluations required to find a solution of fitness of 1e-20 for Fros as a function of the pool size (N here lambda).

A parametric study to find the optimal population size.

ResultsResults

Best Fitness versus function evaluations for Schwefel’s function.

ResultsResults

Comparison of G3 PCX with Differential Evolution

Felp Fsch

Frose

ResultsResults

Scale Up study for G3 PCX

Felp Fsch

Frose

ConclusionsConclusions

A comprehensive parametric study of Real Parameter GA s was performed.

Comparisons of PCX with UNDX, SPX, ES, DE and classical methods were performed.

Superiority of Parent Centric Recombination Approach was seen.

Future Work. -- Need to develop a mathematical model for

choosing the best set of parameters for a Real Parameter GA.

-- Replacing traditional recombination operators in Multi-objective GA s with PCX and seeing the effect.