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I am trying to write a program for maximizing a function using a genetic algorithm. The function has $n$ integer variables $x_1 \dots x_n$, such that each variable is in the range [-n,n].

What is the most common way to binary-encode integers in the range [-n, n]?

Will it work if, for the initial population, instead of generating numbers between -n and n, I generate numbers from 0 to 2n and encode them in binary and then, while decoding, I subtract n every time for every subsequent generation ?

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@Renan: I am sorry i forgot to mention that I want to use binary encoding. So I am wondering about a good way to binary-encode integers in the range [-n,n]. I have modified the question. – vjain27 Apr 23 '13 at 1:58
Deleted my comment since it's not relevant, then. – Renan Apr 23 '13 at 1:58

The most common way is called two's compliment ("'s_complement"), which is essentially the bitwise not of the original number expressed in binary plus one, where the first bit represents the negativity of the number. Alternatively, you could just add another bit to the binary representation which signals whether the number is positive or negative.

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I guess these would make crossover more problematic. Especially the sign-bit representation. For e.g. if during crossover a sign-bit position is selected as the crossover point. – vjain27 Apr 23 '13 at 2:08
I think that the GA should converge to giving low fitness values to elements with the wrong sign bit. As you suggested, the problem is somewhat less with two's compliment, as flipping a bit leads to a change of at most $2^{n-1}$ (n is the number of bits), whereas using the sign bit would allow the value to change by $2^n$ if the sign bit was flipped. – Micah Apr 23 '13 at 2:12

The most common binary representation that I have seen in GA's is grey codes ( In this case, you would represent the numbers as $[0,2n]$, and you would probably have an extra bit in there if $n$ was a power of 2. No matter what, there's no method of binary representation that can span $[-n,n]$ without having some unused representations.

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