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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The basic genetic algorithm

• 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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The basic genetic algorithm

• 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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Generation based GAs

Then create twochildren from eachparent.

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Generation based GAs

Thenreplace theoriginalpopulationby thechildren

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Generation based GAs

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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Steady-State based GAs

Select twoparents. Thencreate a child.

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Steady-State based GAs

Then replace amember of thepopulation withthe new child.

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Steady-State based GAs

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

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Find the city with next smallest key,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.