1 简介
针对布谷鸟仿生智能优化算法存在着的易陷入局部最优、求解精度低、以及收敛速度慢等问题，本文提出了基于多阶段动态扰动和动态惯性权重的布谷鸟搜索算法(MACS)。首先，利用多阶段动态扰动策略对布谷鸟算法的全局位置的最优鸟巢位置根据方差可调的正态随机分布进行扰动，有利于增加种群的多样性和鸟窝位置的灵活性，提高算法全局搜索能力。其次，在局部位置处引入动态惯性权重，使得算法有效克服易陷入局部最优的缺陷，提高局部寻优搜索能力。最后，引入了动态切换概率p代替固定概率，可以动态平衡全局搜索和局部搜索。通过与 4 种算法相比和 11 个测试函数的仿真结果表明：改进布谷鸟算法(MACS)的寻优性能明显提高，收敛速度更快，求解精度更高，具有更强的全局搜索能力和跳出局部最优能力。
2 部分代码
%% 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 randomization
new_nest=empty_nests(nest,Lb,Ub,pa) ;
% Evaluate this set of solutions
for i=1:n,
%% Evalute the fitness/function values of the new population
f_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 solutions
X((n+1):(2*n),:)=[nest f]; % Combine old solutions
Sorted=solutions_sorting(X, m, d);
%% Select n solutions from a combined population of 2*n solutions
new_Sol=Select_pop(Sorted, m, d, n);
% Decompose the sorted solutions into solutions, fitness & ranking
nest=new_Sol(:,1:d); % Sorted solutions/variables
f=new_Sol(:,(d+1):(d+m)); % Sorted objective values
RnD=new_Sol(:,(d+m+1):end); % Sorted ranks and distances
%% Running display at each 100 iterations
if ~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 algorithm
u=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 flights
s=s+stepsize.*randn(size(s));
% Apply simple bounds/limits
nest(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 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;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 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 solutions
RankSol = max(find(sorted_nest(:, Krank) == i));
% In the new cuckoos/solutions, there can be npop solutions to fill
if 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 out
if 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),:);
end
end
% Record and update the current rank after adding new cuckoo solutions
K = RankSol;
end
3 仿真结果
4 参考文献
[1]薛益鸽, and 邓辉文. "基于动态分组与高斯扰动的改进布谷鸟算法." 重庆师范大学学报（自然科学版） 035.002(2018):108-113.
