15.053 tuesday, may 14

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15.053 Tuesday, May 14

• Genetic Algorithms

Handouts: Lecture Notes

Question: when should there be an additional review session?

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The basic genetic algorithm

• Developed by John Holland in 1975

• Simulates the process of evolution

• Basic Principle: Evolution can be viewed as an optimizing process

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More on physical analogies

• Physical Analogies as a guiding principle for optimization problems – Genetic Algorithms • John Holland 1975 – Simulated Annealing • Kirkpatrick – Ant Colony Systems

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• Loosely modeled on natural selection with a touch of molecular biology thrown in.

• Fitter individuals mate [selection operator].

• The chromosomes of each child are formed as a mixture of the chromosomes of the parents [crossover operator].

• Mutation adds diversity within the species and a greater scope for improvement [mutation operator].

• Chromosomes encode the relevant information.

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GA terms

chromosome(solution)

gene

(variable)

alleles

(values)

selection

crossover

mutation

population Objective: maximize fitness function

(objective function)

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Selection Operator:

Selects two parents from the population for mating.The selection is biased towards fitter individuals.

Crossover Operator:

Each child is obtained as a random mixture of itsparents using a crossover operation.

Mutation Operator:

At times an individual in the population undergoes a random mutation.

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A Simple Example: Maximize the number of 1’s

• Initial Population Fitness

• 1 1 1 0 1 4• 0 1 1 0 1 3• 0 0 1 1 0 2• 1 0 0 1 1 3

• Average fitness 3 Usually populations are much bigger, say around 50 to 100, or more.

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Crossover Operation: takes two solutions andcreates a child (or more) whose genes are amixture of the genes of the parents.

parent 1 parent 2

0 1 1 0 1 1 0 0 1 1

Select two parentsfrom thepopulation.

This is theselection step.There will bemore on this later.

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Crossover Operation: takes two solutions andcreates a child (or more) whose genes are amixture of the genes of the parents.

parent 1 parent 2

0 1 1 0 1 1 0 0 1 1

1 point crossover: Divide each parent into twoparts at the same location k (chosen randomly.)

Child 1 consists of genes 1 to k-1 from parent 1and genes k to n from parent 2. Child 2 is the“reverse”.

child 1 child 2 0 1 1 1 1 1 0 0 0 1

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Selection Operator

• Think of crossover as mating• Selection biases mating so that fitter parents are more likely to mate.

For example, let the probability of selectingmember j be fitness(j)/total fitness

Example:

1. 1 1 1 0 1 4

2. 0 1 1 0 1 3

3. 0 0 1 1 0 2

4. 1 0 0 1 1 3

Total fitness 12

Prob(1) = 4/12 = 1/3

Prob(3) = 2/12 = 1/6

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Example with Selection and Crossover Only

original after 5 after 10 generations generations

1 0 0 1 1 1 1 0 1 1 1 1 0 1 10 1 0 0 0 1 0 1 1 1 1 0 0 1 10 0 0 0 1 1 1 0 1 1 1 1 0 1 11 1 1 1 1 1 1 0 1 1 1 1 0 1 1. . .0 0 1 0 0 1 1 0 1 1 1 1 0 1 11 1 0 1 1 1 1 0 1 1 1 1 0 1 1

2.8000 3.7000 3.9000

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Mutation

• Previous difficulty: important genetic variability was lost from the population

• Idea: introduce genetic variability into the population through mutation

• simple mutation operation: randomly flip q% of the alleles (bits) in the population.

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Previous Example with a 1% mutation rate

original after 5 after 10 generations generations

1 0 0 1 1 1 1 0 1 1 1 1 1 1 10 1 0 0 0 1 1 1 1 1 1 1 1 1 10 0 0 0 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1. . .1 0 0 0 1 0 1 1 1 1 1 1 1 1 10 0 1 0 0 1 1 1 1 1 1 1 1 1 11 1 0 1 1 1 1 1 1 1 1 1 1 1 1

2.8000 4.8000 4.9000

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Representations of Operators

illustrated on a bit-based representation

Selection Crossover Mutation

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• define the representation – how the solution is represented

• define “fitness” function – objective function

• define the operators – initialization, crossover, mutation

initialize population

select two parents

create 1 or 2children

mutate population

modify population

should we stop orgo on?

finish

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Generation based GAs

In generation basedGA’s we createchildren onegeneration at a time.

Take the entirepopulation of n andcreate n/2 sets ofparents usingselection.

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Then create twochildren from eachparent.

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Thenreplace theoriginalpopulationby thechildren

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This creates the nextgeneration. Theniterate

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Steady-State based GAs

In steady state GA’swe create one child atat time, and thenreplace one memberof the population withthe new child.

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Select twoparents. Thencreate a child.

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Then replace amember of thepopulation withthe new child.

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Repeat thisprocess, andoccasionallyform amutation.

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Steady-State Genetic Algorithm

begin obtain initial population repeat select two individuals I1 and I2; apply the crossover operator on I1 and I2

to produce a child I3; replace a member of the population with I3 or discard I3; (often a parent is replaced) occasionally perform a mutation or an immigration; until the population converges;end;

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Encoding Schemes (These can be tricky forcombinatorial problems)

• How does one encode a tour on n cities?

• Representation 1: the cities in order

• Representation 2: the list of “next cities”

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How does one do a crossover?

• It’s very difficult in this case. It’s not clear how to mix two tours.

• People in the GA community have often relied on ad hoc methods, and have not been so successful.

You can’t take half of one tour and half of another

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How does one do a crossover?

• Standard rule: find something that works. – Example: visit the first k cities in the order that they appear on the first tour, and then visit the remaining cities in the order that they appear on the second tour.

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Random Keys Representation• A chromosome consists of n integers from 0 to K – in our example, K = 100• By sorting the n integers, one can obtain an order

Here are the “random keys” for cities 1 to 9

This is the order obtained by taking cities inorder of their random keys.

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A crossover using random keys

parent 1

parent 2

Select a key from one of the two parent (randomly)

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parent 1

parent 2

child

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An alternative crossover that relies ona randomized algorithm

• The random insertion algorithm Choose three cities randomly and obtain a tour T on the cities For k = 4 to n, choose a city that is not on T and insert it optimally into T.

• Representation: use random keys. In other words select cities to insert in the order that they appear in random keys

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Choose the 3 cities with smallest keys,and create a tour on these cities.

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Find the city with next smallestkey, and insert it.

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Find the city with next smallest key,and insert it.

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Insert the 8th city

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Insert the last city

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The tour associated with the randomkeys using insertion.

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GAs Nbhd Search• Population• Fitness function• Mutation Operator (operates on single solutions)• Crossover Operator (creates random child from two parents)• Selection operator (bias towards fitter individuals)• Important use of randomization

• One solution at a time• Objective function

• Neighborhood operator

• Local search: always look for an improvement.

• Usually does not rely on randomization

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GAs since 1975

• GAs started with 1-point crossover, bit flipping for mutation, and a generational scheme.

• GAs have evolved a lot since then – Lots of approaches for representations, for mutations, for crossover, for combining multiple heuristics, … – It’s no longer obvious what is or is not a genetic algorithm.

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Features of Genetic Algorithms

• GAs seem to work best in problems in which the function is very complex, and feasibility is easy to achieve.

• GA implementation is still an art. They often require a lot of tweaking, then again sometimes you can tweak all you want and they will still find the same result.

• Easy to make parallel

• Easy to get started and get an approach that is working

• Often requires lots of effort to make it work well.

15.053 tuesday, may 14

Documents

parents crossover operator

crossover operation

random mutation

improvement mutation

mutation rateoriginal

crossover onlyoriginal

point crossover

fitter parents