Why have a crossover value that isn't 0.5?

Question

Most of the literature I've read about GAs suggests using a crossover value of around 0.7, so you take the first 70% of one chromosome's genes, and the last 30% of the other to produce one new chromosome.

If you are picking the parent chromosomes by taking the top two (ranked by fitness), then I can see the logic here, as you are giving more weight to the genes of the higher-rated chromosome. However, if you're using a stochastic method (such as a roulette wheel) to pick the parents, then what's the point of using anything other than 0.5 as the crossover value? Given that you have picked chromosomes A and B as the parents, you're just as likely to pick A first and B second as B first and A second aren't you?

I've only actually written one GA so far (still way down at the bottom of the learning curve, but moving up fast thanks to some great help here!), but experiments on that show that 0.5 gives a faster convergence to the solution than any other value.

Or am I missing something?

Answer 1

The ideal crossover operation depends very much on the problem space. The underlying assumptions of evolutionary and genetic algorithms is that two good solutions can sometimes be combined into a better solution – good solutions look similar to other good solutions. This intuitively makes sense if the problem space has a single optimum to which each successive generation will converge.

When there are multiple optima, the space in between these optima is by definition not optimal. If we take a chromosome A that is near one optimum and combine it with a chromosome B from near another optimum, we'll land in between, and will likely have a resulting chromosome c that is worse than its parents. Staying closer to one or the other parents increases the likelihood of getting a chromosome d that is better or at least not much worse than the parents.

     _                d         ^ fitness
    / \              d \        |
   /   A            B   \       |
__/     \___ccc___dd     \____  |
-----------------------------------> chromosome space
     |                |
     |     valley     |
     |     of "meh"   |
1. optimum         2. optimum

The crossover value is just one algorithm parameter you can tune to suit your problem structure. Sometimes you'll see faster convergence with a low crossover value, sometimes with a very high crossover value. But for very high values, this would be less like a crossover but only a very little change like a mutation. So instead of using a value near 1.0, you'd rather reduce the crossover rate and increase the mutation rate.

Answer 2

Confusingly, crossover rate and mutation rate are, while being named similarly, typically interpreted differently.

Mutation rate of x% ==> You perform the mutation operator with probability 1.0, and each application of that operator will change x% of the bits of the mutated individual.

Crossover rate of x% ==> You choose to perform crossover at all with probability x.

So a crossover rate of 70% doesn't mean you take 70% of the bits from parent 1 and 30% from parent 2. It means that you'll perform whatever crossover operator you have chosen 70% of the time. The remaining 30% of the time, you'll pass the parents unmodified into the offspring pool.