Use MATLAB to thoroughly understand the principle of genetic algorithm + code explanation + specific examples

Initialize the population

Follow the examples to learn the genetic algorithm, such as calculating the optimal value of y=xsin(3 x), first look at the code

​
clear;clc;close all;

%%遗传参数设置
NUMPOP=100;%初始种群大小
irange_l=-1; %问题解区间
irange_r=2;
LENGTH=22; %二进制编码长度
ITERATION = 10000;%迭代次数
CROSSOVERRATE = 0.7;%杂交率
SELECTRATE = 0.5;%选择率
VARIATIONRATE = 0.001;%变异率

​

Here we need to set the parameters of the genetic parameters. The size of the initial population can be changed, such as 100, 1000, 2000, etc. The solution interval of the problem is the value range of the independent variable corresponding to the function, and the length of the binary code can also be changed. The number of iterations, hybridization rate, selection rate, and mutation rate are all given values.

%初始化种群
pop=m_InitPop(NUMPOP,irange_l,irange_r);
pop_save=pop;
%绘制初始种群分布
x=linspace(-1,2,1000);
y=m_Fx(x);
plot(x,y);
hold on
for i=1:size(pop,2)
    plot(pop(i),m_Fx(pop(i)),'ro');
end
hold off
title('初始种群');

The next step is to initialize the population. For m_InitPop and m_Fx, let's take a look at the specific function form:

function pop=m_InitPop(numpop,irange_l,irange_r)
%% 初始化种群
%  输入：numpop--种群大小；
%       [irange_l,irange_r]--初始种群所在的区间
pop=[];
for i=1:numpop
    pop(:,i)=irange_l+(irange_r-irange_l)*rand;
end
end
    

 pop(:,i)=irange_l+(irange_r-irange_l)*rand; This we generate an initial population in the interval of the independent variable, and this population can also set its own data.

function y=m_Fx(x)
%% 要求解的函数
    y=x.*sin(3*pi.*x);
end

The image of the initial population is:

start iterating

Our aim is to find the optimal value of y=xsin(3 x) according to the genetic algorithm.

%开始迭代
for time=1:ITERATION
    %计算初始种群的适应度
    fitness=m_Fitness(pop);
    %选择
    pop=m_Select(fitness,pop,SELECTRATE);
    %编码
    binpop=m_Coding(pop,LENGTH,irange_l);
    %交叉
    kidsPop = crossover(binpop,NUMPOP,CROSSOVERRATE);
    %变异
    kidsPop = Variation(kidsPop,VARIATIONRATE);
    %解码
    kidsPop=m_Incoding(kidsPop,irange_l);
    %更新种群
    pop=[pop kidsPop];
end

Then start iterating, look at the iteration diagram

 adaptability

Let's see how to calculate the fitness, selection, crossover, and mutation of the population, and explain them in sub-functions

    %计算初始种群的适应度
    fitness=m_Fitness(pop);

We calculate the fitness of the population according to the m_Fitness function:

function fitness=m_Fitness(pop)
%% Fitness Function
%y=xsin(3x)在[-1,2]上，最大值也不会超过2
%所以计算函数值到2的距离，距离最小时，即为最优解
%适应度函数为1/距离
for n=1:size(pop,2)
    fitness(n)=1/(2-m_Fx(pop(:,n)));
end

end

 Input the initial population pop, and output the fitness corresponding to each individual, fitness(n)=1/(2-m_Fx(pop(:,n))); here you can add a small value such as esp. Prevent the denominator from appearing 0.

choose

    %选择
    pop=m_Select(fitness,pop,SELECTRATE);

According to the selection rate of SELECTRATE = 0.5, after generating the initial population solution, firstly, according to the selection rate, for example, 50% of the individuals are selected for crossover and mutation process. Special attention should be paid to the fact that the selection rate (also called the generation gap) cannot be set to 100%. Although the optimal value is recorded in each cycle, the optimal value can be obtained faster, but it violates the principle of the algorithm.

function parentPop=m_Select(matrixFitness,pop,SELECTRATE)
%% 选择
% 输入：matrixFitness--适应度矩阵
%      pop--初始种群
%      SELECTRATE--选择率

sumFitness=sum(matrixFitness(:));%计算所有种群的适应度

accP=cumsum(matrixFitness/sumFitness);%累积概率
%轮盘赌选择算法
for n=1:round(SELECTRATE*size(pop,2))
    matrix=find(accP>rand); %找到比随机数大的累积概率
    if isempty(matrix)
        continue
    end
    parentPop(:,n)=pop(:,matrix(1));%将首个比随机数大的累积概率的位置的个体遗传下去
end
end

On the basis of evaluating the fitness of the individual, the optimized individual is directly inherited to the next generation through the selection operation, or a new individual is generated through paired crossover and then passed on to the next generation. Let the group size be size(pop,2), and the probability of an individual being selected be accP.

coding

    %编码
    binpop=m_Coding(pop,LENGTH,irange_l);

The main function is to convert decimal to binary

function binPop=m_Coding(pop,pop_length,irange_l)
%% 二进制编码（生成染色体）
% 输入：pop--种群
%      pop_length--编码长度
pop=round((pop-irange_l)*10^6);
for n=1:size(pop,2) %列循环
    for k=1:size(pop,1) %行循环
        dec2binpop{k,n}=dec2bin(pop(k,n));%dec2bin的输出为字符向量；
                                          %dec2binpop是cell数组
        lengthpop=length(dec2binpop{k,n});
        for s=1:pop_length-lengthpop %补零
            dec2binpop{k,n}=['0' dec2binpop{k,n}];
        end
    end
    binPop{n}=dec2binpop{k,n};   %取dec2binpop的第k行
end

    

cross

Interleave after converting to binary:

    %交叉
    kidsPop = crossover(binpop,NUMPOP,CROSSOVERRATE);

%% 子函数
%
%题  目：Crossover
%
%%
%输   入：
%           parentsPop       上一代种群
%           NUMPOP           种群大小
%           CROSSOVERRATE    交叉率
%输   出：
%           kidsPop          下一代种群
%
%% 
function kidsPop = Crossover(parentsPop,NUMPOP,CROSSOVERRATE)
kidsPop = {[]};n = 1;
while size(kidsPop,2)<NUMPOP-size(parentsPop,2)
    %选择出交叉的父代和母代
    father = parentsPop{1,ceil((size(parentsPop,2)-1)*rand)+1};
    mother = parentsPop{1,ceil((size(parentsPop,2)-1)*rand)+1};
    %随机产生交叉位置
    crossLocation = ceil((length(father)-1)*rand)+1;
    %如果随即数比交叉率低，就杂交
    if rand<CROSSOVERRATE
        father(1,crossLocation:end) = mother(1,crossLocation:end);
        kidsPop{n} = father;
        n = n+1;
    end
end

Mutations

I won't explain too much here, the code can be used directly. Next is mutation

    %变异
    kidsPop = Variation(kidsPop,VARIATIONRATE);

%% 子函数
%
%题  目：Variation
%
%
%输   入：
%           pop              种群
%           VARIATIONRATE    变异率
%输   出：
%           pop              变异后的种群
%% 
function kidsPop = Variation(kidsPop,VARIATIONRATE)
for n=1:size(kidsPop,2)
    if rand<VARIATIONRATE
        temp = kidsPop{n};
        %找到变异位置
        location = ceil(length(temp)*rand);
        temp = [temp(1:location-1) num2str(~temp(location))...
            temp(location+1:end)];
       kidsPop{n} = temp;
    end
end

decoding

Decoding is performed after mutation, that is, binary is converted to decimal.

    %解码
    kidsPop=m_Incoding(kidsPop,irange_l);

function pop=m_Incoding(binPop,irange_l)
%% 解码
popNum=1;
popNum = 1;%染色体包含的参数数量
for n=1:size(binPop,2)
    Matrix = binPop{1,n};
    for num=1:popNum
        pop(num,n) = bin2dec(Matrix);
    end
end
pop = pop./10^6+irange_l;

update population

Then update the population,

    %更新种群
    pop=[pop kidsPop];

Finally, draw the termination population after the iteration of the genetic algorithm:

figure
x=linspace(-1,2,1000);
y=m_Fx(x);
plot(x,y);
hold on
for i=1:size(pop,2)
    plot(pop(i),m_Fx(pop(i)),'ro');
end
hold off
title('终止种群');

terminate population

You can see the optimal solution for the image:

 We can also output the optimal solution and maximum fitness:

disp(['最优解：' num2str(max(m_Fx(pop)))]);
disp(['最大适应度：' num2str(max(m_Fitness(pop)))]);   

output:

最优解：1.491
最大适应度：1.9645

This is for the optimization of one-dimensional functions, and the optimization of two-dimensional functions will continue to be explained in future articles.