（一）白鲸优化（BWO）算法
白鲸优化（BWO）算法一种基于群体的元启发式算法，用于解决优化问题。BWO的灵感来自白鲸的行为，包括三个阶段：探索阶段，开发阶段和鲸鱼坠落阶段。
（二）Matlab代码
1.主函数
BWO
function [xposbest,fvalbest,Curve] = BWO(Npop,Max_it,lb,ub,nD,fobj)
%NPOP-临时种群
% Max_it-最大迭代次数
%lb-下限
%ub-上限
%nD-维度
%fobj-适应度函数
% disp('Beluga Whale Optimization is optimizing your problem');
fit = inf*ones(Npop,1);
newfit = fit;
Curve = inf*ones(1,Max_it);
kk = zeros(1,Max_it);
Counts_run = 0;
if size(ub,2)==1
lb = lb*ones(1,nD); ub = ub*ones(1,nD);
end
pos = rand(Npop,nD).*(ub-lb)+lb;
for i = 1:Npop
fit(i,1) = fobj(pos(i,:));
Counts_run = Counts_run+1;
end
[fvalbest,index]=min(fit);
xposbest = pos(index,:);
T = 1;
while T <= Max_it
newpos = pos;
WF = 0.1-0.05*(T/Max_it); % The probability of whale fall
kk = (1-0.5*T/Max_it)*rand(Npop,1); % The probability in exploration or exploitation
for i = 1:Npop
if kk(i) > 0.5 % exploration phase
r1 = rand(); r2 = rand();
RJ = ceil(Npop*rand); % Roulette Wheel Selection
while RJ == i
RJ = ceil(Npop*rand);
end
if nD <= Npop/5
params = randperm(nD,2);
newpos(i,params(1)) = pos(i,params(1))+(pos(RJ,params(1))-pos(i,params(2)))*(r1+1)*sin(r2*360);
newpos(i,params(2)) = pos(i,params(2))+(pos(RJ,params(1))-pos(i,params(2)))*(r1+1)*cos(r2*360);
else
params=randperm(nD);
for j = 1:floor(nD/2)
newpos(i,2*j-1) = pos(i,params(2*j-1))+(pos(RJ,params(1))-pos(i,params(2*j-1)))*(r1+1)*sin(r2*360);
newpos(i,2*j) = pos(i,params(2*j))+(pos(RJ,params(1))-pos(i,params(2*j)))*(r1+1)*cos(r2*360);
end
end
else % exploitation phase
r3 = rand(); r4 = rand(); C1 = 2*r4*(1-T/Max_it);
RJ = ceil(Npop*rand); % Roulette Wheel Selection
while RJ == i
RJ = ceil(Npop*rand);
end
alpha=3/2;
sigma=(gamma(1+alpha)*sin(pi*alpha/2)/(gamma((1+alpha)/2)*alpha*2^((alpha-1)/2)))^(1/alpha); % Levy flight
u=randn(1,nD).*sigma;
v=randn(1,nD);
S=u./abs(v).^(1/alpha);
KD = 0.05;
LevyFlight=KD.*S;
newpos(i,:) = r3*xposbest - r4*pos(i,:) + C1*LevyFlight.*(pos(RJ,:)-pos(i,:));
end
% boundary
Flag4ub = newpos(i,:)>ub;
Flag4lb = newpos(i,:)<lb;
newpos(i,:)=(newpos(i,:).*(~(Flag4ub+Flag4lb)))+ub.*Flag4ub+lb.*Flag4lb;
newfit(i,1) = fobj(newpos(i,:)); % fitness calculation
Counts_run = Counts_run+1;
if newfit(i,1) < fit(i,1)
pos(i,:) = newpos(i,:);
fit(i,1) = newfit(i,1);
end
end
for i = 1:Npop
% whale falls
if kk(i) <= WF
RJ = ceil(Npop*rand); r5 = rand(); r6 = rand(); r7 = rand();
C2 = 2*Npop*WF;
stepsize2 = r7*(ub-lb)*exp(-C2*T/Max_it);
newpos(i,:) = (r5*pos(i,:) - r6*pos(RJ,:)) + stepsize2;
% boundary
Flag4ub = newpos(i,:)>ub;
Flag4lb = newpos(i,:)<lb;
newpos(i,:)=(newpos(i,:).*(~(Flag4ub+Flag4lb)))+ub.*Flag4ub+lb.*Flag4lb;
newfit(i,1) = fobj(newpos(i,:)); % fitness calculation
Counts_run = Counts_run+1;
if newfit(i,1) < fit(i,1)
pos(i,:) = newpos(i,:);
fit(i,1) = newfit(i,1);
end
end
end
[fval,index]=min(fit);
if fval<fvalbest
fvalbest = fval;
xposbest = pos(index,:);
end
kk_Record(T) = kk(1);
Curve(T) = fvalbest;
T = T+1;
end
% display(['The function call is ', num2str(Counts_run)]);
2.测试函数
Get_Functions_details
function [lb,ub,dim,fobj] = Get_Functions_details(F)
switch F
case 'F1'
fobj = @F1;
lb=-100;
ub=100;
dim=30;
case 'F2'
fobj = @F2;
lb=-10;
ub=10;
dim=30;
case 'F3'
fobj = @F3;
lb = -1;
ub = 1;
dim = 30;
case 'F4'
fobj = @F4;
lb=-100;
ub=100;
dim=30;
case 'F5'
fobj = @F5;
lb=-100;
ub=100;
dim=30;
case 'F6'
fobj = @F6;
lb=-30;
ub=30;
dim=30;
case 'F7'
fobj = @F7;
lb=-100;
ub=100;
dim=30;
case 'F8'
fobj = @F8;
lb=-1.28;
ub=1.28;
dim=30;
case 'F9'
fobj = @F9;
lb=-5;
ub=10;
dim=30;
case 'F10'
fobj = @F10;
lb=-500;
ub=500;
dim=30;
case 'F11'
fobj = @F11;
lb=-10;
ub=10;
dim=30;
case 'F12'
fobj = @F12;
lb=-5;
ub=5;
dim=30;
case 'F13'
fobj = @F13;
lb=-5.12;
ub=5.12;
dim=30;
case 'F14'
fobj = @F14;
lb=-32;
ub=32;
dim=30;
case 'F15'
fobj = @F15;
lb=-600;
ub=600;
dim=30;
case 'F16'
fobj = @F16;
lb=-10;
ub=10;
dim=30;
case 'F17'
fobj = @F17;
lb=-50;
ub=50;
dim=30;
case 'F18'
fobj = @F18;
lb=-50;
ub=50;
dim=30;
case 'F19'
fobj = @F19;
lb=-65;
ub=65;
dim=2;
case 'F20'
fobj = @F20;
lb=-5;
ub=5;
dim=4;
case 'F21'
fobj = @F21;
lb=-5;
ub=5;
dim=2;
case 'F22'
fobj = @F22;
lb=0;
ub=10;
dim=4;
case 'F23'
fobj = @F23;
lb=0;
ub=10;
dim=4;
case 'F24'
fobj = @F24;
lb=0;
ub=10;
dim=4;
end
end
% F1
function o = F1(x)
o=sum(x.^2);
end
% F2
function o = F2(x)
o=sum(abs(x))+prod(abs(x));
end
% F3
function o = F3(x)
dim = size(x,2);
o=0;
for i=1:dim
o=o+abs(x(i))^(i+1);
end
end
% F4
function o = F4(x)
dim=size(x,2);
o=0;
for i=1:dim
o=o+sum(x(1:i))^2;
end
end
% F5
function o = F5(x)
o=max(abs(x));
end
% F6
function o = F6(x)
dim=size(x,2);
o=sum(100*(x(2:dim)-(x(1:dim-1).^2)).^2+(x(1:dim-1)-1).^2);
end
% F7
function o = F7(x)
o=sum(abs((x+.5)).^2);
end
% F8
function o = F8(x)
dim=size(x,2);
o=sum([1:dim].*(x.^4))+rand;
end
% F9
function o = F9(x)
dim = size(x,2);
o = sum(x.^2)+(sum(0.5*[1:dim].*x))^2+(sum(0.5*[1:dim].*x))^4;
end
% F10
function o = F10(x)
o=sum(-x.*sin(sqrt(abs(x))));
end
% F11
function o = F11(x)
dim = size(x,2);
o = 1+sum(sin(x(1:dim)).^2)-exp(-sum(x.^2));
end
% F12
function o = F12(x)
dim = size(x,2);
o = 0.5*sum(x(1:dim).^4-16*x(1:dim).^2+5*x(1:dim));
end
% F13
function o = F13(x)
dim=size(x,2);
o=sum(x.^2-10*cos(2*pi.*x))+10*dim;
end
% F14
function o = F14(x)
dim=size(x,2);
o=-20*exp(-.2*sqrt(sum(x.^2)/dim))-exp(sum(cos(2*pi.*x))/dim)+20+exp(1);
end
% F15
function o = F15(x)
dim=size(x,2);
o=sum(x.^2)/4000-prod(cos(x./sqrt([1:dim])))+1;
end
% F16
function o = F16(x)
dim = size(x,2);
o = (sum(sin(x(1:dim)).^2) - exp(-sum(x.^2)))*exp(-sum(sin(sqrt(abs(x(1:dim)))).^2));
end
% F17
function o = F17(x)
dim=size(x,2);
o=(pi/dim)*(10*((sin(pi*(1+(x(1)+1)/4)))^2)+sum((((x(1:dim-1)+1)./4).^2).*...
(1+10.*((sin(pi.*(1+(x(2:dim)+1)./4)))).^2))+((x(dim)+1)/4)^2)+sum(Ufun(x,10,100,4));
end
% F18
function o = F18(x)
dim=size(x,2);
o=.1*((sin(3*pi*x(1)))^2+sum((x(1:dim-1)-1).^2.*(1+(sin(3.*pi.*x(2:dim))).^2))+...
((x(dim)-1)^2)*(1+(sin(2*pi*x(dim)))^2))+sum(Ufun(x,5,100,4));
end
% F19
function o = F19(x)
aS=[-32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32;,...
-32 -32 -32 -32 -32 -16 -16 -16 -16 -16 0 0 0 0 0 16 16 16 16 16 32 32 32 32 32];
for j=1:25
bS(j)=sum((x'-aS(:,j)).^6);
end
o=(1/500+sum(1./([1:25]+bS))).^(-1);
end
% F20
function o = F20(x)
aK=[.1957 .1947 .1735 .16 .0844 .0627 .0456 .0342 .0323 .0235 .0246];
bK=[.25 .5 1 2 4 6 8 10 12 14 16];bK=1./bK;
o=sum((aK-((x(1).*(bK.^2+x(2).*bK))./(bK.^2+x(3).*bK+x(4)))).^2);
end
% F21
function o = F21(x)
o=4*(x(1)^2)-2.1*(x(1)^4)+(x(1)^6)/3+x(1)*x(2)-4*(x(2)^2)+4*(x(2)^4);
end
% F22
function o = F22(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:5
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
% F23
function o = F23(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:7
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
% F24
function o = F24(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:10
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
3.主程序
main
clear all; close all; clc; Function_name = 'F1'; % function name Npop = 50; % Number of search agents Max_it = 1000; % Maximum number of iterations [lb,ub,nD,fobj]=Get_Functions_details(Function_name); [xposbest,fvalbest,Curve]=BWO(Npop,Max_it,lb,ub,nD,fobj);
引用：Zhong Changting (2023). Beluga whale optimization (BWO) (https://www.mathworks.com/matlabcentral/fileexchange/112830-beluga-whale-optimization-bwo) MATLAB Central File Exchange.