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Genetic Algorithms

Pre-lecture quiz

Genetic Algorithms (GA) are based on evolutionary approach to AI, in which methods of evolution of population is used to obtain an optimal solution for a given problem. They were proposed in 1975 by John Henry Holland.

Genetic Algorithms are based on the following ideas:

Valid solutions to the problem can be represented as genes
Crossover allows us to combine two solutions together to obtain new valid solution
Selection is used to select more optimal solutions using some fitness function
Mutations are introduced to destabilize optimization and get us out of the local minimum

If you want to implement a Genetic Algorithm, you need the following:

To find a method of coding our problem solutions using genes g∈Γ
On the set of genes Γ we need to define fitness function fit: Γ→R. Smaller function values correspond to better solutions.
To define crossover mechanism to combine two genes together to get a new valid solution crossover: Γ^2→Γ.
To define mutation mechanism mutate: Γ→Γ. In many cases, crossover and mutation are quite simple algorithms to manipulate genes as numeric sequences or bit vectors.

Specific implementation of a genetic algorithm can vary from case to case, but overall structure is the following:

Select initial population G⊂Γ
Randomly select one of the operations that will be performed at this step: crossover or mutation
Crossover:

Randomly select two genes g₁, g₂ ∈ G
Compute crossover g=crossover(g₁,g₂)
If fit(g)<fit(g₁) or fit(g)<fit(g₂) - replace corresponding gene in the population by g.

Mutation - select random gene g∈G and replace it by mutate(g)
Repeat from step 2, until we get sufficiently small value of fit, or until the limit on the number of steps is reached.

Typical Tasks

Tasks typically solved by GA:

Schedule optimization
Optimal packing
Optimal cutting
Speeding up exhaustive search

Notebooks

Go to Genetic.ipynb notebooks to see two examples of using Genetic Algorithms:

Fair division of treasure
8 Queen Problem

Post-lecture quiz

Assignment

Your goal is to solve so-called Diophantine equation - an equation with integer roots. For example, consider the equation a+2b+3c+4d=30. You need to find integer roots that satisfy this equation.

Hints:

You can consider roots to be in the interval [0;30]
As a gene, consider using the list of root values

Use Diophantine.ipynb as a starting point.

This assignment is inspired by this post.

✅ Todo: conclusion, challenge, reference.

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