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I recently learned about Genetic algorithms and I solved the 8 queens problem using a genetic algorithm but I don't know how to optimize any functions using a genetic algorithm.

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I want a guide on how to find chromosomes and fitness function for such a function? And I don't want code.

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Yours is a really nice and easy question: you've seen how to use GAs in a complex problem but you're missing how to apply them to the most basic of all. I'll show you:

In a real-world implementation we have to structure our problem to be solved with GAs; we need to modify it, finding an equivalent representation that accepts individuals, each one of these is built from a genome and must be evaluated using a fitness function.

If you see this graphically you'll discover that the fitness function is just describing an area (or line or volume, depending on the grade) on our space and we are randomly dropping individuals on it, awarding the ones that fell higher than the others. We then try to modify the genomes of these guys to move them towards peaks of this fitness function. Crossover vs mutation, I did this image, feel free to use it.

In practice the fitness function is our world, our ground truth and we are exploring it.

Now to the function optimization part: well, we do not need any abstraction here, any strange individual representation or transformation of the problem; we just want to find a maximum, and this is exactly what GAs are for!

So, let's have a look at the elements you need to solve your problem with GAs:

  • Fitness function => you have it already! It's the function you want to find the max of.
  • Individual => as the maximum is enough, so the individual will be just a point in space.
  • Genome => like every respective point in space, the genome will just be a collection of real numbers (one for each dimension).

Now, like in any GA, you will instantiate randomly an initial population (random points in the domain). The evaluation of course is the easy part, just put them into the function and you'll se the value of the individuals.
What about crossover and mutation though? You'll have to use techniques that work with real-number genes, like BLXalpha, BLXalphabeta. These are just defining ranges between values and picking random values inside these ranges, I wrote a pretty detailed answer about this, you can check it out at: https://ai.stackexchange.com/a/6323/15530

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