matlab实现遗传算法解决优化问题_用遗传算法工具箱或编程实现遗传算法求解下面的函数优化问题

作者：AllinToyou | 2024-05-08 13:45:10

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用遗传算法工具箱或编程实现遗传算法求解下面的函数优化问题

引言

最近刚学完遗传算法，觉得遗传算法比较重要，因此我就手动编写对其进行实现，在此记录一下。编写遗传算法是为了解决下面的问题：
$\min f\left( x_1,x_2 \right) =20+{x_1}^2+{x_2}^2-10\left( \cos 2\pi x_1+\cos 2\pi x_2 \right) \\ s.t. -5<x_1,x_2<5$

遗传算法原理

遗传算法是模拟生物种群进化的原理而设计出来的算法，是一种随机全局搜索优化方法，它模拟了自然选择和遗传中发生的复制、交叉(crossover)和变异(mutation)等现象，从任一初始种群（Population）出发，通过随机选择、交叉和变异操作，产生一群更适合环境的个体，使群体进化到搜索空间中越来越好的区域，这样一代一代不断繁衍进化，最后收敛到一群最适应环境的个体（Individual），从而求得问题的优质解。值得说明的是，遗传算法相对较好的解决了由神经网络中梯度法带来的陷入局部极小值的问题，是一个很好的算法。

matlab代码实现

% 时间：2023.5.31
% 开发者：落霞孤鹜，秋水长天
% 代码涉及算法：基本遗传算法(SGA)
% 代码功能:使用SGA解决以下问题：
% min f(x1,x2)=20+x1^2+x2^2-10(cos(2*pi*x1)+cos(2*pi*x2))  s.t. -5<x1,x2<5

%% 主函数
clc;
clear all;
close all;
global population1 popSize populationAll crossProbality mutationProbality generationTimes generationNow
global maxFitness fitnessValue1 stringLength trueValue averageFitness trueValueForMaxFitness
popSize = 100;                                  %种群规模
stringLength = 20;                              %编码长度，小数点后五位精度，最少需要20位
population1 = ones(popSize,stringLength,2);     %种群1
trueValue = zeros(popSize,2);
populationAll = ones(popSize,stringLength,2,popSize);        %迭代过程种群汇总
fitnessValue1 = zeros(popSize,1);                %每代种群适应度
crossProbality = 0.6;                                %交叉概率
mutationProbality = 0.03;                             %变异概率
generationTimes = 500;                             %迭代代数
generationNow = 1;
maxFitness = zeros(generationTimes,1);
averageFitness = zeros(generationTimes,1);
trueValueForMaxFitness = zeros(generationTimes,2);
randn('seed',sum(100*clock))
initPopulation;
fitnessCalculate;
select;
while generationNow<generationTimes+1
    generation;  %遗传操作
    generationNow = generationNow+1;
end
showResult;

%% 初始化种群
function initPopulation()
    global population1 popSize populationAll stringLength 
    for i1 = 1:popSize
        for i2 = 1:stringLength
            population1(i1,i2,1) = int8(rand()>0.5);    %随机生成数据
            population1(i1,i2,2) = int8(rand()>0.5);
        end
    end
    populationAll(:,:,:,1) = population1;
end


%% 遗传函数
function generation()
    fitnessCalculate;                  %计算适应度值及排序
    select;                         %选择操作
    crossover;                      %交叉
    mutation;                       %变异
end


%% 适应度计算
function fitnessCalculate()
    global popSize fitnessValue1 population1 stringLength trueValue averageFitness 
    global generationNow maxFitness trueValueForMaxFitness
    format long
    x = zeros(popSize,2);
    dataNeedDecoding = zeros(2,stringLength);
    averageFitness(generationNow) = 0;
    maxFitness(generationNow) = 0;
    for i = 1:popSize
        dataNeedDecoding(1,:) = population1(i,:,1)';
        dataNeedDecoding(2,:) = population1(i,:,2)';
        x(i,:) = decoding(dataNeedDecoding)';
        if x(i,1)>5||x(i,1)<-5||x(i,2)>5||x(i,2)<-5
            fitnessValue1(i) =0;
            trueValue(i,:) = x(i,:);
        else
            trueValue(i,:) = x(i,:);
            fitnessValue1(i) = 90-(20 + x(i,1)^2 + x(i,2)^2 - 10*(cos(2*3.14*x(i,1))+cos(2*3.14*x(i,2))));
        end
        averageFitness(generationNow) = averageFitness(generationNow) + fitnessValue1(i);
        if i == 1
            maxFitness(generationNow) = fitnessValue1(i);
            trueValueForMaxFitness(generationNow,:) = trueValue(i,:);
        else
            if maxFitness(generationNow)<fitnessValue1(i)
                maxFitness(generationNow) = fitnessValue1(i);
                trueValueForMaxFitness(generationNow,:) = trueValue(i,:);
            end
        end
    end
    averageFitness(generationNow) = averageFitness(generationNow) / 1.0 / popSize;
end



%% 选择算子
function select()
    format long
    global popSize fitnessValue1  population1 stringLength
    wheelForAll = zeros(popSize,1);
    fitnessValueAll = 0;
    population_temp = ones(popSize,stringLength,2);
    for i1 = 1:popSize
        fitnessValueAll = fitnessValueAll + fitnessValue1(i1);
    end
    for i2 = 1:popSize
        temp = fitnessValue1(i2)*1.0/fitnessValueAll;
        if i2 == 1
            wheelForAll(i2) = temp;
        else
            wheelForAll(i2) = wheelForAll(i2-1) + temp;
        end
    end
    for i3 = 1:popSize                                               %轮盘赌法进行选择
        a = rand();
        for i4 = 1:popSize
            if a<wheelForAll(i4)
                population_temp(i3,:,:) = population1(i4,:,:);
                break;
            end
        end
    end
    population1 = population_temp;
end


%% 交叉算子
function crossover()
    global popSize population1 stringLength crossProbality
    for i1 = 1:2:popSize
        a = rand();
        if a<crossProbality
            crossPosition = randi([1 stringLength]);
            tempBlock = population1(i1,1:crossPosition,:);
            population1(i1,1:crossPosition,:) = population1(i1+1,1:crossPosition,:);           % 单点交叉
            population1(i1+1,1:crossPosition,:) = tempBlock;
        end
    end
end


%% 变异算子
function mutation()
    global popSize mutationProbality population1 stringLength
    for i1 = 1:popSize
        a = rand();
        if a<mutationProbality
            mutationPosition = randi([1 stringLength]);
            population1(i1,mutationPosition,:) = int8((population1(i1,mutationPosition,:)+1)/2);
        end
    end
end



%% 解码
function [decimalismData1,decimalismData2] = decoding(binaryData)
    binaryData1 = num2str(binaryData(1,:));
    binaryData2 = num2str(binaryData(2,:));
    decimalismData1 = bin2dec(binaryData1);
    decimalismData2 = bin2dec(binaryData2);
    decimalismData1 = decimalismData1 /100000.0 - 5;
    decimalismData2 = decimalismData2 /100000.0 - 5;
end


%% 显示数据
function showResult()
    global averageFitness maxFitness trueValueForMaxFitness generationTimes
    figure(1)
    i = 1:generationTimes;
    plot(i,averageFitness,'b')
    hold on
    plot(i,maxFitness,'r')
    hold off
    legend('平均适应度变化','最大适应度变化')
    title('遗传算法迭代相关参数变化')
    figure(2)
    v = -5:0.01:5;
    a = length(v);
    x1 = zeros(a,a);
    x2 = zeros(a,a);
    y = zeros(a,a);
    for i1 = 1:a
        for i2 = 1:a
            x1(i1,i2) = v(i1);
            x2(i1,i2) = v(i2);
            y(i1,i2) = 20 + x1(i1,i2)^2 + x2(i1,i2).^2 - 10*(cos(2*3.14*x1(i1,i2))+cos(2*3.14*x2(i1,i2)));
        end
    end
    surf(x1,x2,y,'EdgeColor', 'none');
    hold on
    a1 = trueValueForMaxFitness(generationTimes,1)
    a2 = trueValueForMaxFitness(generationTimes,2)
    y1 = 20 + a1^2 + a2^2 - 10*(cos(2*3.14*a1)+cos(2*3.14*a2))
    plot3(a1,a2,y1,'r*');
    hold off
end
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实验结果

这里研究的函数图像为：
在这里插入图片描述
通过遗传算法，其平均适应度和最大适应度变化为：

最终优化结果在函数图像上的位置：

结语

不得不说明的是，遗传算法有一定的概率性，因此，得到的不一定是最优解，只能说是相对较优的解。

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