1 簡介

針對布谷鳥仿生智能優化算法存在著的易陷入局部最優、求解精度低、以及收斂速度慢等問題，本文提出了基于多階段動態擾動和動態慣性權重的布谷鳥搜索算法(MACS)。首先，利用多階段動態擾動策略對布谷鳥算法的全局位置的最優鳥巢位置根據方差可調的正態隨機分布進行擾動，有利于增加種群的多樣性和鳥窩位置的靈活性，提高算法全局搜索能力。其次，在局部位置處引入動態慣性權重，使得算法有效克服易陷入局部最優的缺陷，提高局部尋優搜索能力。最后，引入了動態切換概率p代替固定概率，可以動態平衡全局搜索和局部搜索。通過與 4 種算法相比和 11 個測試函數的仿真結果表明：改進布谷鳥算法(MACS)的尋優性能明顯提高，收斂速度更快，求解精度更高，具有更強的全局搜索能力和跳出局部最優能力。

2 部分代碼

%% 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) ? ?% % Multiobjective cuckoo search (MOCS) added in July 2012, ? ? ? ? % % Then, MOCS was updated in Sept 2015. ? ? ? ? ? ? ? ? ? ? Thanks % % ----------------------------------------------------------------- %% References -- 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 %% 3) X.-S. Yang, S. Deb, Multi-objective cuckoo search for? % Design optimization, Computers & Operations Research,? % vol. 40, no. 6, 1616-1624 (2013). % ----------------------------------------------------------------% % 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 Optimization Algoirthms, ? ? ? ?%? % ? ? Elsevier Press, 2014. ?" ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?% % --------------------------------------------------------------- %% =============================================================== % %% Notes: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? % % 1) The constraint-handling is not included in this demo code. ? % % The main idea to show how the essential steps of cuckoo search ?% % and multi-objective cuckoo search (MOCS) can be done. ? ? ? ? ? % % 2) Different implementations may lead to slightly different ? ? % % behavour and/or results, but there is nothing wrong with it, ? ?% % as it is the nature of random walks and all metaheuristics. ? ? % % --------------------------------------------------------------- % function [bestnest,fmin]=mocs_new(inp) if nargin<1, inp=[100 1000 0.25]; % pop_size, #iteraion, pa end ? ? % Number of nests (or the population size) n=inp(1); % Number of iterations/generations N_IterTotal=inp(2); % Discovery rate of alien eggs/solutions pa=inp(3); d=30; ? % Dimensionality of the problem % Simple lower bounds Lb=0*ones(1,d);? % Simple upper bounds Ub=1*ones(1,d);% Number of objectives m=2;%% Initialize the population for i=1:n,Sol(i,:)=Lb+(Ub-Lb).*rand(1,d);?f(i,1:m) = obj_funs(Sol(i,:), m); end % Store the fitness or objective values f_new=f; %% Sort the initialized population x=[Sol f]; ?% combined into a single input % Non-dominated sorting for the initila population Sorted=solutions_sorting(x, m,d); % Decompose into solutions, fitness, rank and distances nest=Sorted(:,1:d); f=Sorted(:,(d+1):(d+m)); RnD=Sorted(:,(d+m+1):end);%% Starting iterations for t=1:N_IterTotal,% Generate new solutions (but keep the current best)new_nest=get_cuckoos(nest,nest(1,:), Lb,Ub); ??% ? new_nest=nest;% Discovery and randomizationnew_nest=empty_nests(nest,Lb,Ub,pa) ;% Evaluate this set of solutionsfor i=1:n,%% Evalute the fitness/function values of the new populationf_new(i, 1:m) = obj_funs(new_nest(i,1:d),m);if (f_new(i,1:m) <= f(i,1:m)), ?f(i,1:m)=f_new(i,1:m);nest(i,:)=new_nest(i,:);end% Update the current best (stored in the first row)if (f_new(i,1:m) <= f(1,1:m)),?nest(1,1:d) = new_nest(i,1:d);?f(1,:)=f_new(i,:);end ? ? ? ??end ?% end of for loop%% Combined population consits of both the old and new solutions %% So the total number of solutions for sorting is 2*n %% ! It's very important to combine both populations, otherwise, %% the results may look odd and will be very inefficient. !X(1:n,:)=[new_nest f_new]; ? ? ?% Combine new solutionsX((n+1):(2*n),:)=[nest f]; ? ? ?% Combine old solutionsSorted=solutions_sorting(X, m, d);?%% Select n solutions from a combined population of 2*n solutionsnew_Sol=Select_pop(Sorted, m, d, n);% Decompose the sorted solutions into solutions, fitness & rankingnest=new_Sol(:,1:d); ? ? ? ? ? % Sorted solutions/variablesf=new_Sol(:,(d+1):(d+m)); ? ? ?% Sorted objective valuesRnD=new_Sol(:,(d+m+1):end); ? ?% Sorted ranks and distances%% Running display at each 100 iterationsif ~mod(t,100),?disp(strcat('Iterations t=',num2str(t)));?plot(f(:, 1), f(:, 2),'rs','MarkerSize',3);?axis([0 1 -0.8 1]);xlabel('f_1'); ylabel('f_2');drawnow;end ??end %% End of iterations%% --------------- All subfunctions are list below ------------------ ? ? % %% Get cuckoos by ramdom walk function nest=get_cuckoos(nest,best,Lb,Ub) n=size(nest,1); % For details, please see the chapters of the following Elsevier book: ? % X. S. Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014). beta=3/2; ?% Levy exponent in Levy flights 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,:);%% Levy flights by Mantegna's algorithmu=randn(size(s))*sigma;v=randn(size(s));step=u./abs(v).^(1/beta);stepsize=0.1*step.*(s-best);% Now the actual random walks or flightss=s+stepsize.*randn(size(s));% Apply simple bounds/limitsnest(j,:)=simplebounds(s,Lb,Ub); end%% 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,d]=size(nest); % The solutions represented by cuckoos to be discovered or not? % with a probability pa. This action is implemented as a status vector K=rand(size(nest))>pa;? %% New solution by biased/selective random walks stepsize=rand(1,d).*(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 bounds function s=simplebounds(s,Lb,Ub)% Apply the lower boundns_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;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Objective functions? function f = obj_funs(x, m) % Zitzler-Deb-Thiele's funciton No 3 (ZDT function 3) % M = # of objectives % d = # of variables/dimensions d=length(x); ?% d=30 for ZDT 3 % First objective f1 f(1) = x(1); g=1+9/29*sum(x(2:d)); h=1-sqrt(f(1)/g)-f(1)/g*sin(10*pi*f(1)); % Second objective f2 f(2) = g*h; %%%%%%%%%%%%%%%%%% end of the definitions of obojectives %%%%%%%%%%%%%%%%%%function new_Sol = Select_pop(nest, m, ndim, npop) % The input population to this part has twice (ntwice) of the needed? % population size (npop). Thus, selection is done based on ranking and? % crowding distances, calculated from the non-dominated sorting ntwice= size(nest,1); % Ranking is stored in column Krank Krank=m+ndim+1; % Sort the population of size 2*npop according to their ranks [~,Index] = sort(nest(:,Krank)); sorted_nest=nest(Index,:); % Get the maximum rank among the population RankMax=max(nest(:,Krank));?%% Main loop for selecting solutions based on ranks and crowding distances K = 0; ?% Initialization for the rank counter? % Loop over all ranks in the population for i =1:RankMax, ?% Obtain the current rank i from sorted solutionsRankSol = max(find(sorted_nest(:, Krank) == i));% In the new cuckoos/solutions, there can be npop solutions to fillif RankSol<npop,new_Sol(K+1:RankSol,:)=sorted_nest(K+1:RankSol,:);end?% If the population after addition is large than npop, re-arrangement% or selection is carried outif RankSol>=npop% Sort/Select the solutions with the current rank?candidate_nest = sorted_nest(K + 1 : RankSol, :);[~,tmp_Rank]=sort(candidate_nest(:,Krank+1),'descend');% Fill the rest (npop-K) cuckoo/solutions up to npop solutions?for j = 1:(npop-K),?new_Sol(K+j,:)=candidate_nest(tmp_Rank(j),:);endend% Record and update the current rank after adding new cuckoo solutionsK = RankSol; end

3 仿真結果

4 參考文獻

[1]薛益鴿, and 鄧輝文. "基于動態分組與高斯擾動的改進布谷鳥算法." 重慶師范大學學報（自然科學版） 035.002(2018):108-113.?

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