多目标粒子群优化，MATLAB代码直接复制，适合新手！

该算法简单易懂，很适合刚入门多目标算法的。且想改进其他单目标优化算法为多目标的，完全可以在此算法框架上直接修改！今天重新推出一下。

多目标粒子群算法是应用最广泛，也是最经典的多目标寻优算法。各种硕士博士文章，都将其应用在各种各样的领域。今天就为大家带来一期多目标粒子群算法。

与网上大多数多目标粒子群代码不同，本期给出的多目标粒子群优化算法，只有一个脚本和一个函数，很适合新手学习，而且出图精美！

在经典的多目标测试函数“ZDT1”,“ZDT2”,“ZDT3”,“ZDT6”，“Kursawe”，“Schaffer”，“Poloni”，“Viennet2”，“Viennet3”中对多目标粒子群进行测试，结果如下：

其中绿色的线代表真实的Pareto前沿面，黑色的圆圈表示寻优得到的Pareto值，红色的圈表示其他粒子。

ZDT1

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ZDT2

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ZDT3

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ZDT6

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Kursawe

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Schaffer

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Viennet2

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Viennet3

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可以看到，在这几个经典函数中的测试，多目标粒子群的效果还是非常不错的，但也有可改进的空间。

接下来直接上代码！

首先是主函数：

clear all; clc;% Multi-objective function% MultiObjFnc = 'Schaffer';% MultiObjFnc = 'Kursawe';% MultiObjFnc = 'Poloni';% MultiObjFnc = 'Viennet2';% MultiObjFnc = 'Viennet3';% MultiObjFnc = 'ZDT1';% MultiObjFnc = 'ZDT2';% MultiObjFnc = 'ZDT3';% MultiObjFnc = 'ZDT6';switch MultiObjFnc case 'Schaffer' % Schaffer MultiObj.fun = @(x) [x(:).^2, (x(:)-2).^2]; MultiObj.nVar = 1; MultiObj.var_min = -5; MultiObj.var_max = 5; load('Schaffer.mat'); MultiObj.truePF = PF; case 'Kursawe' % Kursawe MultiObj.fun = @(x) [-10.*(exp(-0.2.*sqrt(x(:,1).^2+x(:,2).^2)) + exp(-0.2.*sqrt(x(:,2).^2+x(:,3).^2))), ... sum(abs(x).^0.8 + 5.*sin(x.^3),2)]; MultiObj.nVar = 3; MultiObj.var_min = -5.*ones(1,MultiObj.nVar); MultiObj.var_max = 5.*ones(1,MultiObj.nVar); load('Kursawe.mat'); MultiObj.truePF = PF; case 'Poloni' % Poloni's two-objective A1 = 0.5*sin(1)-2*cos(1)+sin(2)-1.5*cos(2); A2 = 1.5*sin(1)-cos(1)+2*sin(2)-0.5*cos(2); B1 = @(x,y) 0.5.*sin(x)-2.*cos(x)+sin(y)-1.5.*cos(y); B2 = @(x,y) 1.5.*sin(x)-cos(x)+2.*sin(y)-0.5.*cos(y); f1 = @(x,y) 1+(A1-B1(x,y)).^2+(A2-B2(x,y)).^2; f2 = @(x,y) (x+3).^2+(y+1).^2; MultiObj.fun = @(x) [f1(x(:,1),x(:,2)), f2(x(:,1),x(:,2))]; MultiObj.nVar = 2; MultiObj.var_min = -pi.*ones(1,MultiObj.nVar); MultiObj.var_max = pi.*ones(1,MultiObj.nVar); case 'Viennet2' % Viennet2 f1 = @(x,y) 0.5.*(x-2).^2+(1/13).*(y+1).^2+3; f2 = @(x,y) (1/36).*(x+y-3).^2+(1/8).*(-x+y+2).^2-17; f3 = @(x,y) (1/175).*(x+2.*y-1).^2+(1/17).*(2.*y-x).^2-13; MultiObj.fun = @(x) [f1(x(:,1),x(:,2)), f2(x(:,1),x(:,2)), f3(x(:,1),x(:,2))]; MultiObj.nVar = 2; MultiObj.var_min = [-4, -4]; MultiObj.var_max = [4, 4]; load('Viennet2.mat'); MultiObj.truePF = PF; case 'Viennet3' % Viennet3 f1 = @(x,y) 0.5.*(x.^2+y.^2)+sin(x.^2+y.^2); f2 = @(x,y) (1/8).*(3.*x-2.*y+4).^2 + (1/27).*(x-y+1).^2 +15; f3 = @(x,y) (1./(x.^2+y.^2+1))-1.1.*exp(-(x.^2+y.^2)); MultiObj.fun = @(x) [f1(x(:,1),x(:,2)), f2(x(:,1),x(:,2)), f3(x(:,1),x(:,2))]; MultiObj.nVar = 2; MultiObj.var_min = [-3, -10]; MultiObj.var_max = [10, 3]; load('Viennet3.mat'); MultiObj.truePF = PF; case 'ZDT1' % ZDT1 (convex) g = @(x) 1+9.*sum(x(:,2:end),2)./(size(x,2)-1); MultiObj.fun = @(x) [x(:,1), g(x).*(1-sqrt(x(:,1)./g(x)))]; MultiObj.nVar = 30; MultiObj.var_min = zeros(1,MultiObj.nVar); MultiObj.var_max = ones(1,MultiObj.nVar); load('ZDT1.mat'); MultiObj.truePF = PF; case 'ZDT2' % ZDT2 (non-convex) f = @(x) x(:,1); g = @(x) 1+9.*sum(x(:,2:end),2)./(size(x,2)-1); h = @(x) 1-(f(x)./g(x)).^2; MultiObj.fun = @(x) [f(x), g(x).*h(x)]; MultiObj.nVar = 30; MultiObj.var_min = zeros(1,MultiObj.nVar); MultiObj.var_max = ones(1,MultiObj.nVar); load('ZDT2.mat'); MultiObj.truePF = PF; case 'ZDT3' % ZDT3 (discrete) f = @(x) x(:,1); g = @(x) 1+(9/size(x,2)-1).*sum(x(:,2:end),2); h = @(x) 1 - sqrt(f(x)./g(x)) - (f(x)./g(x)).*sin(10.*pi.*f(x)); MultiObj.fun = @(x) [f(x), g(x).*h(x)]; MultiObj.nVar = 5; MultiObj.var_min = 0.*ones(1,MultiObj.nVar); MultiObj.var_max = 1.*ones(1,MultiObj.nVar); load('ZDT3.mat'); MultiObj.truePF = PF; case 'ZDT6' % ZDT6 (non-uniform) f = @(x) 1 - exp(-4.*x(:,1)).*sin(6.*pi.*x(:,1)); g = @(x) 1 + 9.*(sum(x(:,2:end),2)./(size(x,2)-1)).^0.25; h = @(x) 1 - (f(x)./g(x)).^2; MultiObj.fun = @(x) [f(x), g(x).*h(x)]; MultiObj.nVar = 10; MultiObj.var_min = 0.*ones(1,MultiObj.nVar); MultiObj.var_max = 1.*ones(1,MultiObj.nVar); load('ZDT6.mat'); MultiObj.truePF = PF;end% Parametersparams.Np = 200; % Population sizeparams.Nr = 200; % Repository sizeparams.maxgen = 100; % Maximum number of generationsparams.W = 0.4; % Inertia weightparams.C1 = 2; % Individual confidence factorparams.C2 = 2; % Swarm confidence factorparams.ngrid = 20; % Number of grids in each dimensionparams.maxvel = 5; % Maxmium vel in percentageparams.u_mut = 0.5; % Uniform mutation percentage% MOPSOREP = MOPSO(params,MultiObj);% Display infodisplay('Repository fitness values are stored in REP.pos_fit');display('Repository particles positions are store in REP.pos');

然后是多目标粒子群函数代码：

function REP = MOPSO(params,MultiObj)    % Parameters    Np      = params.Np;    Nr      = params.Nr;    maxgen  = params.maxgen;    W       = params.W;    C1      = params.C1;    C2      = params.C2;    ngrid   = params.ngrid;    maxvel  = params.maxvel;    u_mut   = params.u_mut;    fun     = MultiObj.fun;    nVar    = MultiObj.nVar;    var_min = MultiObj.var_min(:);    var_max = MultiObj.var_max(:);        % Initialization    POS = repmat((var_max-var_min)',Np,1).*rand(Np,nVar) + repmat(var_min',Np,1);    VEL = zeros(Np,nVar);    POS_fit  = fun(POS);    if size(POS,1) ~= size(POS_fit,1)        warning(['The objective function is badly programmed. It is not returning' ...            'a value for each particle, please check it.']);    end    PBEST    = POS;    PBEST_fit= POS_fit;    DOMINATED= checkDomination(POS_fit);    REP.pos  = POS(~DOMINATED,:);    REP.pos_fit = POS_fit(~DOMINATED,:);    REP      = updateGrid(REP,ngrid);    maxvel   = (var_max-var_min).*maxvel./100;    gen      = 1;        % Plotting and verbose    if(size(POS_fit,2)==2)        h_fig = figure(1);        h_par = plot(POS_fit(:,1),POS_fit(:,2),'or'); hold on;        h_rep = plot(REP.pos_fit(:,1),REP.pos_fit(:,2),'ok'); hold on;        try            set(gca,'xtick',REP.hypercube_limits(:,1)','ytick',REP.hypercube_limits(:,2)');            axis([min(REP.hypercube_limits(:,1)) max(REP.hypercube_limits(:,1)) ...                  min(REP.hypercube_limits(:,2)) max(REP.hypercube_limits(:,2))]);            grid on; xlabel('f1'); ylabel('f2');        end        drawnow;    end    if(size(POS_fit,2)==3)        h_fig = figure(1);        h_par = plot3(POS_fit(:,1),POS_fit(:,2),POS_fit(:,3),'or'); hold on;        h_rep = plot3(REP.pos_fit(:,1),REP.pos_fit(:,2),REP.pos_fit(:,3),'ok'); hold on;        try                set(gca,'xtick',REP.hypercube_limits(:,1)','ytick',REP.hypercube_limits(:,2)','ztick',REP.hypercube_limits(:,3)');                axis([min(REP.hypercube_limits(:,1)) max(REP.hypercube_limits(:,1)) ...                      min(REP.hypercube_limits(:,2)) max(REP.hypercube_limits(:,2))]);        end        grid on; xlabel('f1'); ylabel('f2'); zlabel('f3');        drawnow;        axis square;    end    display(['Generation #0 - Repository size: ' num2str(size(REP.pos,1))]);        % Main MPSO loop    stopCondition = false;    while ~stopCondition                % Select leader        h = selectLeader(REP);                % Update speeds and positions        VEL = W.*VEL + C1*rand(Np,nVar).*(PBEST-POS) ...                     + C2*rand(Np,nVar).*(repmat(REP.pos(h,:),Np,1)-POS);        POS = POS + VEL;                % Perform mutation        POS = mutation(POS,gen,maxgen,Np,var_max,var_min,nVar,u_mut);                % Check boundaries        [POS,VEL] = checkBoundaries(POS,VEL,maxvel,var_max,var_min);                       % Evaluate the population        POS_fit = fun(POS);                % Update the repository        REP = updateRepository(REP,POS,POS_fit,ngrid);        if(size(REP.pos,1)>Nr)             REP = deleteFromRepository(REP,size(REP.pos,1)-Nr,ngrid);        end                % Update the best positions found so far for each particle        pos_best = dominates(POS_fit, PBEST_fit);        best_pos = ~dominates(PBEST_fit, POS_fit);        best_pos(rand(Np,1)>=0.5) = 0;        if(sum(pos_best)>1)            PBEST_fit(pos_best,:) = POS_fit(pos_best,:);            PBEST(pos_best,:) = POS(pos_best,:);        end        if(sum(best_pos)>1)            PBEST_fit(best_pos,:) = POS_fit(best_pos,:);            PBEST(best_pos,:) = POS(best_pos,:);        end                % Plotting and verbose        if(size(POS_fit,2)==2)            figure(h_fig); delete(h_par); delete(h_rep);            h_par = plot(POS_fit(:,1),POS_fit(:,2),'or'); hold on;            h_rep = plot(REP.pos_fit(:,1),REP.pos_fit(:,2),'ok'); hold on;            try                set(gca,'xtick',REP.hypercube_limits(:,1)','ytick',REP.hypercube_limits(:,2)');                axis([min(REP.hypercube_limits(:,1)) max(REP.hypercube_limits(:,1)) ...                      min(REP.hypercube_limits(:,2)) max(REP.hypercube_limits(:,2))]);            end            if(isfield(MultiObj,'truePF'))                try delete(h_pf); end                h_pf = plot(MultiObj.truePF(:,1),MultiObj.truePF(:,2),'.','color','g'); hold on;            end            grid on; xlabel('f1'); ylabel('f2');            drawnow;            axis square;        end        if(size(POS_fit,2)==3)            figure(h_fig); delete(h_par); delete(h_rep);             h_par = plot3(POS_fit(:,1),POS_fit(:,2),POS_fit(:,3),'or'); hold on;            h_rep = plot3(REP.pos_fit(:,1),REP.pos_fit(:,2),REP.pos_fit(:,3),'ok'); hold on;            try                set(gca,'xtick',REP.hypercube_limits(:,1)','ytick',REP.hypercube_limits(:,2)','ztick',REP.hypercube_limits(:,3)');                axis([min(REP.hypercube_limits(:,1)) max(REP.hypercube_limits(:,1)) ...                      min(REP.hypercube_limits(:,2)) max(REP.hypercube_limits(:,2)) ...                      min(REP.hypercube_limits(:,3)) max(REP.hypercube_limits(:,3))]);            end            if(isfield(MultiObj,'truePF'))                try delete(h_pf); end                h_pf = plot3(MultiObj.truePF(:,1),MultiObj.truePF(:,2),MultiObj.truePF(:,3),'.','color','g'); hold on;            end            grid on; xlabel('f1'); ylabel('f2'); zlabel('f3');            drawnow;            axis square;        end        display(['Generation #' num2str(gen) ' - Repository size: ' num2str(size(REP.pos,1))]);                % Update generation and check for termination        gen = gen + 1;        if(gen>maxgen), stopCondition = true; end    end    hold off;end% Function that updates the repository given a new population and its% fitnessfunction REP = updateRepository(REP,POS,POS_fit,ngrid)    % Domination between particles    DOMINATED  = checkDomination(POS_fit);    REP.pos    = [REP.pos; POS(~DOMINATED,:)];    REP.pos_fit= [REP.pos_fit; POS_fit(~DOMINATED,:)];    % Domination between nondominated particles and the last repository    DOMINATED  = checkDomination(REP.pos_fit);    REP.pos_fit= REP.pos_fit(~DOMINATED,:);    REP.pos    = REP.pos(~DOMINATED,:);    % Updating the grid    REP        = updateGrid(REP,ngrid);end% Function that corrects the positions and velocities of the particles that% exceed the boundariesfunction [POS,VEL] = checkBoundaries(POS,VEL,maxvel,var_max,var_min)    % Useful matrices    Np = size(POS,1);    MAXLIM   = repmat(var_max(:)',Np,1);    MINLIM   = repmat(var_min(:)',Np,1);    MAXVEL   = repmat(maxvel(:)',Np,1);    MINVEL   = repmat(-maxvel(:)',Np,1);        % Correct positions and velocities    VEL(VEL>MAXVEL) = MAXVEL(VEL>MAXVEL);    VEL(VEL<MINVEL) = MINVEL(VEL<MINVEL);    VEL(POS>MAXLIM) = (-1).*VEL(POS>MAXLIM);    POS(POS>MAXLIM) = MAXLIM(POS>MAXLIM);    VEL(POS<MINLIM) = (-1).*VEL(POS<MINLIM);    POS(POS<MINLIM) = MINLIM(POS<MINLIM);end% Function for checking the domination between the population. It% returns a vector that indicates if each particle is dominated (1) or notfunction dom_vector = checkDomination(fitness)    Np = size(fitness,1);    dom_vector = zeros(Np,1);    all_perm = nchoosek(1:Np,2);    % Possible permutations    all_perm = [all_perm; [all_perm(:,2) all_perm(:,1)]];        d = dominates(fitness(all_perm(:,1),:),fitness(all_perm(:,2),:));    dominated_particles = unique(all_perm(d==1,2));    dom_vector(dominated_particles) = 1;end% Function that returns 1 if x dominates y and 0 otherwisefunction d = dominates(x,y)    d = all(x<=y,2) & any(x<y,2);end% Function that updates the hypercube grid, the hypercube where belongs% each particle and its quality based on the number of particles inside itfunction REP = updateGrid(REP,ngrid)    % Computing the limits of each hypercube    ndim = size(REP.pos_fit,2);    REP.hypercube_limits = zeros(ngrid+1,ndim);    for dim = 1:1:ndim        REP.hypercube_limits(:,dim) = linspace(min(REP.pos_fit(:,dim)),max(REP.pos_fit(:,dim)),ngrid+1)';    end        % Computing where belongs each particle    npar = size(REP.pos_fit,1);    REP.grid_idx = zeros(npar,1);    REP.grid_subidx = zeros(npar,ndim);    for n = 1:1:npar        idnames = [];        for d = 1:1:ndim            REP.grid_subidx(n,d) = find(REP.pos_fit(n,d)<=REP.hypercube_limits(:,d)',1,'first')-1;            if(REP.grid_subidx(n,d)==0), REP.grid_subidx(n,d) = 1; end            idnames = [idnames ',' num2str(REP.grid_subidx(n,d))];        end        REP.grid_idx(n) = eval(['sub2ind(ngrid.*ones(1,ndim)' idnames ');']);    end        % Quality based on the number of particles in each hypercube    REP.quality = zeros(ngrid,2);    ids = unique(REP.grid_idx);    for i = 1:length(ids)        REP.quality(i,1) = ids(i);  % First, the hypercube's identifier        REP.quality(i,2) = 10/sum(REP.grid_idx==ids(i)); % Next, its quality    endend% Function that selects the leader performing a roulette wheel selection% based on the quality of each hypercubefunction selected = selectLeader(REP)    % Roulette wheel    prob    = cumsum(REP.quality(:,2));     % Cumulated probs    sel_hyp = REP.quality(find(rand(1,1)*max(prob)<=prob,1,'first'),1); % Selected hypercube        % Select the index leader as a random selection inside that hypercube    idx      = 1:1:length(REP.grid_idx);    selected = idx(REP.grid_idx==sel_hyp);    selected = selected(randi(length(selected)));end% Function that deletes an excess of particles inside the repository using% crowding distancesfunction REP = deleteFromRepository(REP,n_extra,ngrid)    % Compute the crowding distances    crowding = zeros(size(REP.pos,1),1);    for m = 1:1:size(REP.pos_fit,2)        [m_fit,idx] = sort(REP.pos_fit(:,m),'ascend');        m_up     = [m_fit(2:end); Inf];        m_down   = [Inf; m_fit(1:end-1)];        distance = (m_up-m_down)./(max(m_fit)-min(m_fit));        [~,idx]  = sort(idx,'ascend');        crowding = crowding + distance(idx);    end    crowding(isnan(crowding)) = Inf;        % Delete the extra particles with the smallest crowding distances    [~,del_idx] = sort(crowding,'ascend');    del_idx = del_idx(1:n_extra);    REP.pos(del_idx,:) = [];    REP.pos_fit(del_idx,:) = [];    REP = updateGrid(REP,ngrid); end% Function that performs the mutation of the particles depending on the% current generationfunction POS = mutation(POS,gen,maxgen,Np,var_max,var_min,nVar,u_mut)    % Sub-divide the swarm in three parts [2]    fract     = Np/3 - floor(Np/3);    if(fract<0.5), sub_sizes =[ceil(Np/3) round(Np/3) round(Np/3)];    else           sub_sizes =[round(Np/3) round(Np/3) floor(Np/3)];    end    cum_sizes = cumsum(sub_sizes);        % First part: no mutation    % Second part: uniform mutation    nmut = round(u_mut*sub_sizes(2));    if(nmut>0)        idx = cum_sizes(1) + randperm(sub_sizes(2),nmut);        POS(idx,:) = repmat((var_max-var_min)',nmut,1).*rand(nmut,nVar) + repmat(var_min',nmut,1);    end        % Third part: non-uniform mutation    per_mut = (1-gen/maxgen)^(5*nVar);     % Percentage of mutation    nmut    = round(per_mut*sub_sizes(3));    if(nmut>0)        idx = cum_sizes(2) + randperm(sub_sizes(3),nmut);        POS(idx,:) = repmat((var_max-var_min)',nmut,1).*rand(nmut,nVar) + repmat(var_min',nmut,1);    endend

代码中缺乏各个函数真实的Pareto前沿数据，数据没法直接复制在文中。因此作者打包了一下。

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