今天我要讲的内容是布谷鸟算法，英文叫做Cuckoo search (CS algorithm)。首先还是同样，介绍一下这个算法的英文含义， Cuckoo是布谷鸟的意思，啥是布谷鸟呢，是一种叫做布谷的鸟，o(∩_∩)o ，这种鸟她妈很懒，自己生蛋自己不养，一般把它的宝宝扔到别的种类鸟的鸟巢去。但是呢，当孵化后，遇到聪明的鸟妈妈，一看就知道不是亲生的，直接就被鸟妈妈给杀了。于是这群布谷鸟宝宝为了保命，它们就模仿别的种类的鸟叫，让智商或者情商极低的鸟妈妈误认为是自己的亲宝宝，这样它就活下来了。 Search指的是搜索，这搜索可不是谷歌一下，你就知道。而是搜索最优值，举个简单的例子，y=(x-0.5)^2+1，它的最小值是1，位置是(0.5,1)，我们要搜索的就是这个位置。
现在我们应该清楚它是干嘛的了吧，它就是为了寻找最小值而产生的一种算法，有些好装X的人会说，你傻X啊，最小值不是-2a/b吗，用你找啊? 说的不错，确实是，但是要是我们的函数变成 y=sin(x^3+x^2)+e^cos(x^3)+log(tan(x)+10，你怎么办吶？你解不了，就算你求导数，但是你知道怎么解导数等于0吗？所以我们就得引入先进的东西来求最小值。
为了使大家容易理解，我还是用y=(x-0.5)^2+1来举例子，例如我们有4个布谷鸟蛋(也就是4个x坐标)，鸟妈妈发现不是自己的宝宝的概率是0.25，我们x的取值范围是[0,1]之间，于是我们就可以开始计算了。
目标：求x在[0,1]之内的函数y=(x-0.5)^2+1最小值
(1)初始化x的位置，随机生成4个x坐标，x1=0.4，x2=0.6，x3=0.8，x4=0.3 ——> X=[0.4, 0.6 ,0.8, 0.3]
(2)求出y1~y4，把x1~x4带入函数，求得Y=[1,31, 1.46, 1.69, 1.265]，并选取当前最小值ymin= y4=1.265
(3)开始定出一个y的最大值为Y_global=INF(无穷大)，然后与ymin比较，把Y中最小的位置和值保留，例如Y_global=INF>ymin=1.265，所以令Y_global=1.265
(4)记录Y_global的位置，(0.3，1.265)。
(5)按概率0.25，随机地把X中的值过塞子，选出被发现的蛋。例如第二个蛋被发现x2=0.6，那么他就要随机地变换位子，生成一个随机数，例如0.02，然后把x2=x2+0.02=0.62，之后求出y2=1.4794。那么X就变为了X=[0.4, 0.62 ,0.8, 0.3]，Y=[1,31, 1.4794, 1.69, 1.265]。
(6)进行莱维飞行，这名字听起来挺高大上，说白了，就是把X的位置给随机地改变了。怎么变？有一个公式x=x+alpha*L。
L=S*(X-Y_global)*rand3
S=[rand1*sigma/|rand2|]^(1/beta)
sigma=0.6966
beta=1.5
alpha=0.01
rand1~rand3为正态分布的随机数
然后我们把X=[0.4, 0.6 ,0.8, 0.3]中的x1带入公式，首先随机生成rand1=-1.2371，rand2=-2.1935，rand3=-0.3209，接下来带入公式中，获得x1=0.3985
之后同理计算:
x2=0.6172
x3=0.7889
x4=0.3030
(7)更新矩阵X，X=[0.3985, 0.6172, 0.7889, 0.3030]
(8)计算Y=[1.3092, 1.4766, 1.6751, 1.2661]，并选取当前最小值ymin= y4=1.2661，然后与ymin比较，把Y中最小的位置和值保留，例如Y_global=1.265
(9)返回步骤(5)用更新的X去循环执行，经过多次计算即可获得y的最优值和的最值位置(x,y)
代码：
% -----------------------------------------------------------------
% Cuckoo Search (CS) algorithm by Xin-She Yang and Suash Deb %
% Programmed by Xin-She Yang at Cambridge University %
% Programming dates: Nov to June %
% Last revised: Dec (simplified version for demo only) %
% -----------------------------------------------------------------
% Papers --Citation Details:% 1) X.-S. Yang, S. Deb, Cuckoo search via Levy flights,% in: Proc. of World Congress on Nature &Biologically Inspired% Computing (NaBIC ), December , India,% IEEE Publications, USA, pp. 210-214 ().% /PS_cache/arxiv/pdf/1003/1003.1594v1.pdf
% 2) X.-S. Yang, S. Deb, Engineering optimization by cuckoo search,%Int. J. Mathematical Modelling and Numerical Optimisation,% Vol. 1, No. 4, 330-343 ().% /PS_cache/arxiv/pdf/1005/1005.2908v2.pdf
% ----------------------------------------------------------------%
% This demo program only implements a standard version of %
% Cuckoo Search (CS), as the Levy flights and generation of %
% new solutions may use slightly different methods. %
% The pseudo code was given sequentially (select a cuckoo etc), %
% but the implementation here uses Matlab's vector capability, %
% which results in neater/better codes and shorter running time. %
% This implementation is different and more efficient than the %
% the demo code provided inthe book by% "Yang X. S., Nature-Inspired Metaheuristic Algoirthms, %
% 2nd Edition, Luniver Press, (). "%
% --------------------------------------------------------------- %
% =============================================================== %
% Notes: %
% Different implementations may lead to slightly different %
% behavour and/or results, but there is nothing wrong with it, %
% as this is the nature of random walks and all metaheuristics. %
% -----------------------------------------------------------------
% Additional Note: This version uses a fixed number of generation %
% (not a given tolerance) because many readers asked me to add %
% or implement this option. Thanks.%function [bestnest,fmin]=cuckoo_search_new(n)if nargin<1,%Number of nests (or different solutions)
n=25;
end% Discovery rate of alien eggs/solutions
pa=0.25;%% Change this if you want to getbetter results
N_IterTotal=1000;%%Simple bounds of the search domain%Lower bounds
nd=15;
Lb=-5*ones(1,nd);%Upper bounds
Ub=5*ones(1,nd);%Random initial solutionsfor i=1:n,
nest(i,:)=Lb+(Ub-Lb).*rand(size(Lb));
end%Get the current best
fitness=10^10*ones(n,1);
[fmin,bestnest,nest,fitness]=get_best_nest(nest,nest,fitness);
N_iter=0;%%Starting iterationsfor iter=1:N_IterTotal,% Generate newsolutions (but keep the current best)
new_nest=get_cuckoos(nest,bestnest,Lb,Ub);
[fnew,best,nest,fitness]=get_best_nest(nest,new_nest,fitness);%Update the counter
N_iter=N_iter+n;%Discovery and randomization
new_nest=empty_nests(nest,Lb,Ub,pa) ;% Evaluate this setof solutions
[fnew,best,nest,fitness]=get_best_nest(nest,new_nest,fitness);%Update the counter again
N_iter=N_iter+n;%Find the best objective so farif fnew
fmin=fnew;
bestnest=best;
end
end%%End of iterations%% Post-optimization processing%%Display all the nests
disp(strcat('Total number of iterations=',num2str(N_iter)));
fmin
bestnest%% --------------- All subfunctions are list below ------------------
%%Get cuckoos by ramdom walk
function nest=get_cuckoos(nest,best,Lb,Ub)%Levy flights
n=size(nest,1);%Levy exponent and coefficient% For details, see equation (2.21), Page 16 (chapter 2) of the book% X. S. Yang, Nature-Inspired Metaheuristic Algorithms, 2nd Edition, Luniver Press, ().
beta=3/2;
sigma=(gamma(1+beta)*sin(pi*beta/2)/(gamma((1+beta)/2)*beta*2^((beta-1)/2)))^(1/beta);for j=1:n,
s=nest(j,:);% This isa simple way of implementing Levy flights% For standard random walks, use step=1;%% Levy flights by Mantegna's algorithm
u=randn(size(s))*sigma;
v=randn(size(s));
step=u./abs(v).^(1/beta);% In the next equation, the difference factor (s-best) means that% when the solution isthe best solution, it remains unchanged.
stepsize=0.01*step.*(s-best);% Here the factor 0.01 comes from the fact that L/100should the typical% step size of walks/flights where L isthe typical lenghtscale;% otherwise, Levy flights may become too aggresive/efficient,% which makes new solutions (even) jump outside of the design domain%(and thus wasting evaluations).%Now the actual random walks or flights
s=s+stepsize.*randn(size(s));% Apply simple bounds/limits
nest(j,:)=simplebounds(s,Lb,Ub);
end%%Find the current best nest
function [fmin,best,nest,fitness]=get_best_nest(nest,newnest,fitness)% Evaluating all newsolutionsfor j=1:size(nest,1),
fnew=fobj(newnest(j,:));if fnew<=fitness(j),
fitness(j)=fnew;
nest(j,:)=newnest(j,:);
end
end%Find the current best
[fmin,K]=min(fitness) ;
best=nest(K,:);%% Replace some nests by constructing new solutions/nests
function new_nest=empty_nests(nest,Lb,Ub,pa)%A fraction of worse nests are discovered with a probability pa
n=size(nest,1);% Discovered or not --a status vector
K=rand(size(nest))>pa;% In the real world, if a cuckoo's egg is very similar to a host's eggs, then% this cuckoo's egg is less likely to be discovered, thus the fitness should
% be related to the difference in solutions. Therefore, it isa good idea% to do a random walk ina biased way with some random step sizes.%% New solution by biased/selective random walks
stepsize=rand*(nest(randperm(n),:)-nest(randperm(n),:));
new_nest=nest+stepsize.*K;for j=1:size(new_nest,1)
s=new_nest(j,:);
new_nest(j,:)=simplebounds(s,Lb,Ub);
end%Application of simple constraints
function s=simplebounds(s,Lb,Ub)%Apply the lower bound
ns_tmp=s;
I=ns_tmp
ns_tmp(I)=Lb(I);%Apply the upper bounds
J=ns_tmp>Ub;
ns_tmp(J)=Ub(J);% Update this newmove
s=ns_tmp;%%You can replace the following by your own functions% A d-dimensional objective function
function z=fobj(u)%% d-dimensional sphere function sum_j=1^d (u_j-1)^2.% with a minimum at (1,1, ...., 1);
z=sum((u-1).^2);
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