58 lines
2.8 KiB
Markdown
58 lines
2.8 KiB
Markdown
# Genetic Algorithms
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**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](https://en.wikipedia.org/wiki/John_Henry_Holland).
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Genetic Algorithms are based on the following ideas:
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* Valid solutions to the problem can be represented as **genes**
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* **Crossover** allows us to combine two solutions together to obtain new valid solution
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* **Selection** is used to select more optimal solutions using some **fitness function**
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* **Mutations** are introduced to destabilize optimization and get us out of the local minimum
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If you want to implement a Genetic Algorithm, you need the following:
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* To find a method of coding our problem solutions using **genes** g∈Γ
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* On the set of genes Γ we need to define **fitness function** fit: Γ→**R**. Smaller function values correspond to better solutions.
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* To define **crossover** mechanism to combine two genes together to get a new valid solution crossover: Γ<sup>2</sub>→Γ.
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* To define **mutation** mechanism mutate: Γ→Γ.
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In many cases, crossover and mutation are quite simple algorithms to manipulate genes as numeric sequences or bit vectors.
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Specific implementation of a genetic algorithm can vary from case to case, but overall structure is the following:
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1. Select initial population G⊂Γ
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2. Randomly select one of the operations that will be performed at this step: crossover or mutation
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3. **Crossover**:
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* Randomly select two genes g<sub>1</sub>, g<sub>2</sub> ∈ G
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* Compute crossover g=crossover(g<sub>1</sub>,g<sub>2</sub>)
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* If fit(g)<fit(g<sub>1</sub>) or fit(g)<fit(g<sub>2</sub>) - replace corresponding gene in the population by g.
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4. **Mutation** - select random gene g∈G and replace it by mutate(g)
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5. Repeat from step 2, until we get sufficiently small value of fit, or until the limit on the number of steps is reached.
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## Typical Tasks
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Tasks typically solved by GA:
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1. Schedule optimization
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1. Optimal packing
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1. Optimal cutting
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1. Speeding up exhaustive search
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## Notebooks
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Go to [Genetic.ipynb](Genetic.ipynb) notebooks to see two examples of using Genetic Algorithms:
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1. Fair division of treasure
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1. 8 Queen Problem
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## Assignment
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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.
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Hints:
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1. You can consider roots to be in the interval [0;30]
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1. As a gene, consider using the list of root values
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Use [Diophantine.ipynb](Diophantine.ipynb) as a starting point.
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*This assignment is inspired by [this post](https://habr.com/post/128704/).*
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