? ? ? ?遺傳算法是經(jīng)典的智能算法,?經(jīng)常被用來(lái)求解各種N-P問(wèn)題,?各種非線性函數(shù)的優(yōu)化等,?可以實(shí)現(xiàn)各類模型的非最優(yōu)解優(yōu)化.?遺傳算法穩(wěn)定性比較強(qiáng),?優(yōu)化的效果比較好,?不是特別依賴初值,?尤其對(duì)離散自變量的函數(shù)優(yōu)化是很合適的,?比較容易得到理論最優(yōu)解,?整體的運(yùn)行效率比較好,?對(duì)線性或非線性的約束都很友好.

? ? ? ? 遺傳算法的大體流程如下:

?流程包括:

(1)編碼的設(shè)計(jì)方案,?編碼就是染色體的編碼,?每一個(gè)染色體都與目標(biāo)函數(shù)的一個(gè)解是對(duì)應(yīng)的,?也就是遺傳算法的一個(gè)染色體是目標(biāo)函數(shù)解的一個(gè)單射,?一個(gè)染色體唯一對(duì)應(yīng)一個(gè)解(但反過(guò)來(lái)不一定,?不同的染色體可以對(duì)應(yīng)同樣的一個(gè)解).? 編碼的設(shè)計(jì)方案對(duì)于求目標(biāo)函數(shù)解的可行性和效率是非常重要的,?對(duì)于不同的問(wèn)題模型,?我們需要專門設(shè)計(jì)編碼方案,?比如對(duì)于離散的排序問(wèn)題,?我們可以設(shè)計(jì)一個(gè)自然數(shù)排序編碼 [1,3,2,5,4]這樣的排序編碼,?來(lái)映射排序問(wèn)題的自變量或解,?對(duì)于多維實(shí)數(shù)域的函數(shù)優(yōu)化問(wèn)題,?可以把編碼設(shè)計(jì)為[0.1,0.5,1.2,1.3,2.5]這樣的多維實(shí)數(shù)編碼,直接對(duì)應(yīng)自變量x.? 如何設(shè)計(jì)編碼是各類智能算法永恒的課題.

(2)讀取或設(shè)置問(wèn)題模型的數(shù)據(jù)和參數(shù).

(3)設(shè)置遺傳算法參數(shù),?如種群數(shù),?迭代次數(shù),?交叉率,?變異率.

(4)初始化種群,?一般采用隨機(jī)產(chǎn)生一組染色體作為算法的初始染色體.

(5)開始迭代,?進(jìn)行染色體的變異操作.

(6)對(duì)染色體進(jìn)行交叉操作.

(7)對(duì)染色體進(jìn)行解碼,??這個(gè)過(guò)程就是把染色體映射到解

(8)根據(jù)解計(jì)算目標(biāo)函數(shù)值和約束,,?再把解帶入問(wèn)題模型,得到目標(biāo)函數(shù)值以及是否滿足約束等.

(9)根據(jù)解得到的目標(biāo)函數(shù)對(duì)染色體進(jìn)行選擇,?這個(gè)選擇的操作方法也是比較關(guān)鍵的,?這里面體現(xiàn)了進(jìn)化論的思想,?也即適者生存,?或者適者被選中的概率更大,?久而久之,隨著迭代的進(jìn)行,?適應(yīng)性更好的染色體會(huì)存活下來(lái),?最后得到的是較優(yōu)解,?當(dāng)然如果迭代次數(shù)夠充分,?種群數(shù)夠大,?往往是可以得到理論最優(yōu)解的,?但并不能保證你得到.

(10)輸出結(jié)果,?一般包括迭代曲線和最優(yōu)編碼,最優(yōu)解,?最優(yōu)目標(biāo)函數(shù)等.?進(jìn)而完成了整個(gè)算法的構(gòu)建和問(wèn)題模型的求解.

? ? ? ? 那么如何用MATLAB實(shí)現(xiàn)遺傳算法呢?根據(jù)以上的步驟,? 我們的問(wèn)題模型是:?

其中xi∈[0,1], N=5

? ? ? ?那么我們可直接采用5維實(shí)數(shù)編碼構(gòu)造染色體(見染色體函數(shù)mygenchrome.m) ,?

初始化染色體

N=5,?上下限維[0,1]區(qū)間,? 則一個(gè)合法的染色體可表示為[0.1,0.5,0.6,1.0,0.4],?完全均勻隨機(jī)產(chǎn)生, 見染色體函數(shù)mygenchrome.m.

變異(采用單點(diǎn)變異)

(1) 產(chǎn)生一個(gè)隨機(jī)自然數(shù)r1,r1表示第r1位的基因發(fā)生變異

(2) 采用隨機(jī)變異的方式將第r1位的基因進(jìn)行變異.

舉例:

r1=2, 那么染色體的變異為

[0.1,0.5,0.6,1.0,0.4]→[0.1,0.2,0.6,1.0, 0.4]

交叉

(1) 隨機(jī)選擇兩個(gè)染色體作為父本

(2) 產(chǎn)生2個(gè)隨機(jī)自然數(shù)r1和r2

(3) 將兩個(gè)父本染色體r1至r2之間的基因片段進(jìn)行交換, 得到兩個(gè)子代染色體

舉例:

選擇的兩個(gè)父本染色體

[0.1,0.5,0.6,1.0,0.4] ?[0.1,0.5,0.6,0.70,0.54]

r1=2, r2=4

那么交叉過(guò)程為

[0.1,0.5,0.6,1.0,0.4] ?[0.1,0.5,0.6,0.7,0.54]

交叉后

[0.1,0.5,0.6,0.7, 0.4] ?[0.1,0.5,0.6,1.0, 0.54]

其他目標(biāo)函數(shù)何選擇函數(shù)等較為簡(jiǎn)單,?不再贅述.

? ? ? ? ? MATLAB是設(shè)計(jì)遺傳算法以及其他智能算法的非常好的工具,?下面給大家提供下遺傳算法的完整程序:?

主程序main.m

%% 遺傳算法 優(yōu)化函數(shù)
clc;close all;clear all;warning off;%清除變量
rand('seed', 100);
randn('seed', 100);
format long g;


my_N=10; % 設(shè)定優(yōu)化問(wèn)題維數(shù)

% 設(shè)定遺傳算法參數(shù)
my_popsize=500;% 設(shè)定遺傳算法種群數(shù)
my_maxgen=1000;% 設(shè)定遺傳算法迭代次數(shù)
my_PM=0.1;% 設(shè)定變異概率
my_PC=0.8;% 設(shè)定交叉概率

lb=0*ones(1,my_N);
ub=1*ones(1,my_N);

%% 遺傳算法主程序
my_tracemat=zeros(my_maxgen,2);% 設(shè)定性能跟蹤
my_gen=0;
tic;
Chrom_my=mygenchrome(my_popsize,my_N,lb,ub);% 建立種群
Value_my=mydecodingfun(Chrom_my,my_popsize);% 解碼染色體
[vmin_my,indexmin_my]=min(Value_my);
bestChrom_my=Chrom_my(indexmin_my,:);% 記錄最優(yōu)染色體
bestValue_my=vmin_my;% 記錄的最優(yōu)值

%% 遺傳算法優(yōu)化的主循環(huán)
while my_gen<my_maxgen
    %% 遺傳算子
    FitnV=myranking(Value_my);% 分配適應(yīng)度值
    Chrom_my=myselect('rws',Chrom_my,FitnV,1);% 選擇
    Chrom_my=mymutationga(Chrom_my,my_popsize,my_PM,my_N,lb,ub);% 種群變異,單點(diǎn)變異
    Chrom_my=mycrossga(Chrom_my,my_popsize,my_PC,my_N);% 種群交叉,2點(diǎn)交叉
    Value_my=mydecodingfun(Chrom_my,my_popsize);% 解碼染色體
    
    %% 計(jì)算最優(yōu)
    [vmin_my,indexmin_my]=min(Value_my);
    my_gen=my_gen+1;
    
    %% 記錄最優(yōu)
    if bestValue_my>vmin_my
        bestValue_my=vmin_my;%記錄的最優(yōu)值
        bestChrom_my=Chrom_my(indexmin_my,:);
    end
    my_tracemat(my_gen,2)=mean(Value_my);
    my_tracemat(my_gen,1)=bestValue_my;% 保留最優(yōu)
    
end

disp('算法運(yùn)行時(shí)間');
runtime_ga_my=toc

% 顯示結(jié)果
disp('遺傳算法優(yōu)化得到的最優(yōu)目標(biāo)函數(shù)值');
bestValue_my
disp('遺傳算法優(yōu)化得到的最優(yōu)染色體');
bestChrom_my



figure;
plot(my_tracemat(:,1),'r-','linewidth',1);
hold on;
plot(my_tracemat(:,2),'b-','linewidth',1);
legend({'種群最優(yōu)值','種群均值'},'fontname','宋體');
xlabel('迭代次數(shù)','fontname','宋體');
ylabel('目標(biāo)函數(shù)值','fontname','宋體');
title('遺傳算法迭代曲線','fontname','宋體');

交叉函數(shù) mycrossga.m

function Chrom=mycrossga(Chrom,popsize,PC,N)
% 種群交叉,2點(diǎn)交叉
index100=randperm(popsize);
long1=fix(popsize/2);
set1=index100(1:long1);
set2=index100(long1+1:long1*2);
for i=1:long1
    a=rand;
    if a<PC%小于交叉率開始交叉
        index201=set1(i);
        index202=set2(i);
        R1=Chrom(index201,:);
        R2=Chrom(index202,:);
        rmat=randi([1,N],1,2);%
        r1=min(rmat);
        r2=max(rmat);
        
        S1=R1(r1:r2);%交叉位置的節(jié)點(diǎn)
        S2=R2(r1:r2);%交叉位置的節(jié)點(diǎn)
        R1(r1:r2)=S2;
        R2(r1:r2)=S1;
        %更新染色體
        Chrom(index201,:)=R1;
        Chrom(index202,:)=R2;
    end
end

解碼函數(shù)mydecodingfun.m

function Value = mydecodingfun(Chrom,popsize)
%解碼染色體
Value=zeros(popsize,1);
for i=1:popsize
    x=Chrom(i,:);
    y=myfun(x);
    Value(i,1)=y;
end

目標(biāo)函數(shù)myfun.m

function y=myfun(x)

y=sum((x-0.5).^2);

初始染色體函數(shù)mygenchrome.m

function Chrom=mygenchrome(popsize,N,lb,ub)
% 建立種群
Chrom=zeros(popsize,N);
for i=1:popsize
    for j=1:N
        Chrom(i,j)=lb(j)+(ub(j)-lb(j))*rand(1,1);
    end
end

染色體變異函數(shù)mymutationga.m

function Chrom=mymutationga(Chrom,popsize,PM,N,lb,ub)
% 種群變異,單點(diǎn)變異
for i=1:popsize
    a=rand;
    if a<PM
        r=unidrnd(N);
        Chrom(i,r)=lb(r)+(ub(r)-lb(r))*rand(1,1);
    end
end

?

分配適應(yīng)度函數(shù)

function FitnV = myranking(ObjV, RFun, SUBPOP);

% Identify the vector size (Nind)
   [Nind,ans] = size(ObjV);

   if nargin < 2, RFun = []; end
   if nargin > 1, if isnan(RFun), RFun = []; end, end
   if prod(size(RFun)) == 2,
      if RFun(2) == 1, NonLin = 1;
      elseif RFun(2) == 0, NonLin = 0;
      else error('Parameter for ranking method must be 0 or 1'); end
      RFun = RFun(1);
      if isnan(RFun), RFun = 2; end
   elseif prod(size(RFun)) > 2,
      if prod(size(RFun)) ~= Nind, error('ObjV and RFun disagree'); end
   end

   if nargin < 3, SUBPOP = 1; end
   if nargin > 2,
      if isempty(SUBPOP), SUBPOP = 1;
      elseif isnan(SUBPOP), SUBPOP = 1;
      elseif length(SUBPOP) ~= 1, error('SUBPOP must be a scalar'); end
   end

   if (Nind/SUBPOP) ~= fix(Nind/SUBPOP), error('ObjV and SUBPOP disagree'); end
   Nind = Nind/SUBPOP;  % Compute number of individuals per subpopulation
   
% Check ranking function and use default values if necessary
   if isempty(RFun),
      % linear ranking with selective pressure 2
         RFun = 2*[0:Nind-1]'/(Nind-1);
   elseif prod(size(RFun)) == 1
      if NonLin == 1,
         % non-linear ranking
         if RFun(1) < 1, error('Selective pressure must be greater than 1');
         elseif RFun(1) > Nind-2, error('Selective pressure too big'); end
         Root1 = roots([RFun(1)-Nind [RFun(1)*ones(1,Nind-1)]]);
         RFun = (abs(Root1(1)) * ones(Nind,1)) .^ [(0:Nind-1)'];
         RFun = RFun / sum(RFun) * Nind;
      else
         % linear ranking with SP between 1 and 2
         if (RFun(1) < 1 | RFun(1) > 2),
            error('Selective pressure for linear ranking must be between 1 and 2');
         end
         RFun = 2-RFun + 2*(RFun-1)*[0:Nind-1]'/(Nind-1);
      end
   end;

   FitnV = [];

% loop over all subpopulations
for irun = 1:SUBPOP,
   % Copy objective values of actual subpopulation
      ObjVSub = ObjV((irun-1)*Nind+1:irun*Nind);
   % Sort does not handle NaN values as required. So, find those...
      NaNix = isnan(ObjVSub);
      Validix = find(~NaNix);
   % ... and sort only numeric values (smaller is better).
      [ans,ix] = sort(-ObjVSub(Validix));

   % Now build indexing vector assuming NaN are worse than numbers,
   % (including Inf!)...
      ix = [find(NaNix) ; Validix(ix)];
   % ... and obtain a sorted version of ObjV
      Sorted = ObjVSub(ix);

   % Assign fitness according to RFun.
      i = 1;
      FitnVSub = zeros(Nind,1);
      for j = [find(Sorted(1:Nind-1) ~= Sorted(2:Nind)); Nind]',
         FitnVSub(i:j) = sum(RFun(i:j)) * ones(j-i+1,1) / (j-i+1);
         i =j+1;
      end

   % Finally, return unsorted vector.
      [ans,uix] = sort(ix);
      FitnVSub = FitnVSub(uix);

   % Add FitnVSub to FitnV
      FitnV = [FitnV; FitnVSub];
end

選擇函數(shù)myselect.m

function SelCh = myselect(SEL_F, Chrom, FitnV, GGAP, SUBPOP);

% Check parameter consistency
if nargin < 3, error('Not enough input parameter'); end

% Identify the population size (Nind)
[NindCh,Nvar] = size(Chrom);
[NindF,VarF] = size(FitnV);
if NindCh ~= NindF, error('Chrom and FitnV disagree'); end
if VarF ~= 1, error('FitnV must be a column vector'); end

if nargin < 5, SUBPOP = 1; end
if nargin > 4,
    if isempty(SUBPOP), SUBPOP = 1;
    elseif isnan(SUBPOP), SUBPOP = 1;
    elseif length(SUBPOP) ~= 1, error('SUBPOP must be a scalar'); end
end

if (NindCh/SUBPOP) ~= fix(NindCh/SUBPOP), error('Chrom and SUBPOP disagree'); end
Nind = NindCh/SUBPOP;  % Compute number of individuals per subpopulation

if nargin < 4, GGAP = 1; end
if nargin > 3,
    if isempty(GGAP), GGAP = 1;
    elseif isnan(GGAP), GGAP = 1;
    elseif length(GGAP) ~= 1, error('GGAP must be a scalar');
    elseif (GGAP < 0), error('GGAP must be a scalar bigger than 0'); end
end

% Compute number of new individuals (to select)
NSel=max(floor(Nind*GGAP+.5),2);

% Select individuals from population
SelCh = [];
for irun = 1:SUBPOP,
    FitnVSub = FitnV((irun-1)*Nind+1:irun*Nind);
    ChrIx=feval(SEL_F, FitnVSub, NSel)+(irun-1)*Nind;
    SelCh=[SelCh; Chrom(ChrIx,:)];
end


% End of function

選擇函數(shù)用輪盤賭子函數(shù)rws.m

function NewChrIx = rws(FitnV,Nsel)
% 輪盤賭選擇
[Nind,ans] = size(FitnV);
cumfit  = cumsum(FitnV);
trials = cumfit(Nind) .* rand(Nsel, 1);
Mf = cumfit(:, ones(1, Nsel));
Mt = trials(:, ones(1, Nind))';
[NewChrIx, ans] = find(Mt < Mf & ...
    [ zeros(1, Nsel); Mf(1:Nind-1, :) ] <= Mt);

程序結(jié)果如下:

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