Genetic algorithm for unconstrained single-objective optimization problem.

Maximize the objective function F, given the domain of X and a required percision of 0.0001.

max F(x1,x2) = 21.5 + x1*sin(4*pi*x1) + x2*sin(20*pi*x2)

s.t. -3.0 <= x1 <= 12.1 4.1 <= x2 <= 5.8

• The domain of xj is [aj, bj] and the required precision is four places after the decimal point.
• The precision requirement implies that the range of domain of each variable should be divided into at least (bj - aj) * 10^4 size ranges.
• The required bits (denoted with mj) for a variable is calculated as follows: 2^(mj - 1) < (bj - aj) * 10^4 <= 2^mj - 1
• The mapping from a binary string to a real number for variable xj is completed as follows:xj = aj + decimal(substring j) * (bj - aj) / (2^mj - 1)

• In most practices, a roulette wheel approach is adopted as the selection procedure, which is one of the fitness-proportional selection and can select a new population with respect to the probability distribution based on fitness values.
• The roulette wheel can be constructed with the following steps:

step 1: Calculate the total fitness for the population

step 2: Calculate selection probability pk for each chromosome vk

step 3: Calculate cumulative probability qk for each chromosome vk

step 4: Generate a random number r from the range [0, 1].

step 5: If r <= q1, then select the first chromosome v1; otherwise, select the kth chromosome vk ( 2 <= k <= popSize ) such that qk-1 < r <= qk.

• For Roulette Wheel Selection, due to a low selection pressure, the best mutation probability is around 0.01.

Tournament selection: tournament size = t

Repeat t times: choose a random individual from the pop and remember its fitness

Return the best of these t individuals (BTR)

• If t = 2, it is called Binary Tournament Selection.

• For Binary Tournament Selection, due to a high selection pressure, the best mutation probability is around 0.05.

• Crossover used here is two-cut points method, which random selects two cut points.
• Exchanges the middle or side parts of two parents to generate offspring.

• Alters every bit of genes with a probability equal to the mutation rate.

• Alters every individual with a probability equal to the mutation rate. For each mutated gene, randomly select one bit.