今天我要講的內容是布谷鳥算法，英文叫做Cuckoo search (CS algorithm)。首先還是同樣，介紹一下這個算法的英文含義， Cuckoo是布谷鳥的意思，啥是布谷鳥呢，是一種叫做布谷的鳥，o(∩_∩)o ，這種鳥她媽很懶，自己生蛋自己不養，一般把它的寶寶扔到別的種類鳥的鳥巢去。但是呢，當孵化后，遇到聰明的鳥媽媽，一看就知道不是親生的，直接就被鳥媽媽給殺了。於是這群布谷鳥寶寶為了保命，它們就模仿別的種類的鳥叫，讓智商或者情商極低的鳥媽媽誤認為是自己的親寶寶，這樣它就活下來了。 Search指的是搜索，這搜索可不是谷歌一下，你就知道。而是搜索最優值，舉個簡單的例子，y=(x-0.5)^2+1，它的最小值是1，位置是(0.5,1)，我們要搜索的就是這個位置。
現在我們應該清楚它是干嘛的了吧，它就是為了尋找最小值而產生的一種算法，有些好裝X的人會說，你傻X啊，最小值不是-2a/b嗎，用你找啊? 說的不錯，確實是，但是要是我們的函數變成 y=sin(x^3+x^2)+e^cos(x^3)+log(tan(x)+10，你怎么辦吶？你解不了，就算你求導數，但是你知道怎么解導數等於0嗎？所以我們就得引入先進的東西來求最小值。
為了使大家容易理解，我還是用y=(x-0.5)^2+1來舉例子，例如我們有4個布谷鳥蛋（也就是4個x坐標），鳥媽媽發現不是自己的寶寶的概率是0.25，我們x的取值范圍是[0,1]之間，於是我們就可以開始計算了。
目標：求x在[0,1]之內的函數y=(x-0.5)^2+1最小值
（1）初始化x的位置，隨機生成4個x坐標，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帶入函數，求得Y=[1,31, 1.46, 1.69, 1.265]，並選取當前最小值ymin= y4=1.265
（3）開始定出一個y的最大值為Y_global=INF（無窮大），然后與ymin比較，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）記錄Y_global的位置，（0.3，1.265）。
（5）按概率0.25，隨機地把X中的值過塞子，選出被發現的蛋。例如第二個蛋被發現x2=0.6，那么他就要隨機地變換位子，生成一個隨機數，例如0.02，然后把x2=x2+0.02=0.62，之后求出y2=1.4794。那么X就變為了X=[0.4, 0.62 ,0.8, 0.3]，Y=[1,31, 1.4794, 1.69, 1.265]。
（6）進行萊維飛行，這名字聽起來挺高大上，說白了，就是把X的位置給隨機地改變了。怎么變？有一個公式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為正態分布的隨機數
然后我們把X=[0.4, 0.6 ,0.8, 0.3]中的x1帶入公式，首先隨機生成rand1=-1.2371，rand2=-2.1935，rand3=-0.3209，接下來帶入公式中，獲得x1=0.3985
之后同理計算:
x2=0.6172
x3=0.7889
x4=0.3030
（7）更新矩陣X，X=[0.3985, 0.6172, 0.7889, 0.3030]
（8）計算Y=[1.3092, 1.4766, 1.6751, 1.2661]，並選取當前最小值ymin= y4=1.2661，然后與ymin比較，把Y中最小的位置和值保留，例如Y_global=1.265<ymin=1.2661，所以令Y_global=1.265
（9）返回步驟（5）用更新的X去循環執行，經過多次計算即可獲得y的最優值和的最值位置(x,y)
代碼：
% ----------------------------------------------------------------- % Cuckoo Search (CS) algorithm by Xin-She Yang and Suash Deb % % Programmed by Xin-She Yang at Cambridge University % % Programming dates: Nov 2008 to June 2009 % % Last revised: Dec 2009 (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 2009), December 2009, India, % IEEE Publications, USA, pp. 210-214 (2009). % http://arxiv.org/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 (2010). % http://arxiv.org/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 in the book by % "Yang X. S., Nature-Inspired Metaheuristic Algoirthms, % % 2nd Edition, Luniver Press, (2010). " % % --------------------------------------------------------------- % % =============================================================== % % 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 get better 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 solutions for 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 iterations for iter=1:N_IterTotal, % Generate new solutions (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 set of 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 far if fnew<fmin, 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, (2010). 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 is a 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 is the best solution, it remains unchanged. stepsize=0.01*step.*(s-best); % Here the factor 0.01 comes from the fact that L/100 should the typical % step size of walks/flights where L is the typical lenghtscale; % otherwise, Levy flights may become too aggresive/efficient, % which makes new solutions (even) jump out side 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 new solutions for 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 is a good idea % to do a random walk in a 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<Lb; ns_tmp(I)=Lb(I); % Apply the upper bounds J=ns_tmp>Ub; ns_tmp(J)=Ub(J); % Update this new move 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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