NSGA-II matlab 代码

以下是一个基本的NSGA-II matlab代码，用于解决多目标优化问题。其中包括选择，交叉和变异算子，以及多个目标函数的定义和限制条件的设置。你可以根据你的问题进行修改和适应。

% NSGA-II算法的matlab实现
% 用于解决多目标优化问题
% 参考文献: Deb, Kalyanmoy, et al. "A fast and elitist multiobjective 
% genetic algorithm: NSGA-II." IEEE transactions on evolutionary computation 
% 6.2 (2002): 182-197.

clc;
clear;
close all;

%% 初始化参数

pop_size = 100;             % 种群大小
num_obj = 2;                % 目标函数的个数
max_gen = 500;              % 最大迭代次数
pc = 0.8;                   % 交叉概率
pm = 0.1;                   % 变异概率
eta_c = 20;                 % 交叉分布指数
eta_m = 20;                 % 变异分布指数
l_bound = [-5 -5];          % 变量的下限
u_bound = [5 5];            % 变量的上限

%% 初始化种群
pop = repmat(struct('x',[],'f',[],'rank',[],'dist',[]), pop_size, 1);

for i = 1 : pop_size
    pop(i).x = l_bound + rand(1,num_obj) .* (u_bound - l_bound);
    pop(i).f = evaluate_objective(pop(i).x);
end

%% 进化
for gen = 1 : max_gen
    
    %% 快速非支配排序
    F = fast_non_domination_sort(pop);
    
    %% 计算拥挤度距离
    for i = 1 : length(F)
        pop(F(i).ids) = calculate_crowding_distance(pop(F(i).ids));
    end
    
    %% 选择
    pop = selection(pop, F);
    
    %% 交叉
    pop = crossover(pop, pc, eta_c);
    
    %% 变异
    pop = mutation(pop, pm, eta_m, l_bound, u_bound);
    
    %% 评估
    for i = 1 : pop_size
        pop(i).f = evaluate_objective(pop(i).x);
    end
    
    %% 合并种群
    new_pop = [pop; offspring];
    
    %% 快速非支配排序
    F = fast_non_domination_sort(new_pop);
    
    %% 计算拥挤度距离
    for i = 1 : length(F)
        new_pop(F(i).ids) = calculate_crowding_distance(new_pop(F(i).ids));
    end
    
    %% 选择
    pop = replace(new_pop, pop_size);
    
end

%% 最终结果
F = fast_non_domination_sort(pop);
pareto_front = pop(F(1).ids);
plot(pareto_front.x(:,1),pareto_front.x(:,2),'o');
xlabel('Objective 1');
ylabel('Objective 2');
title('Pareto Front'); 

%% 目标函数
function f = evaluate_objective(x)
    f(1) = x(1)^2 + x(2)^2;
    f(2) = (x(1)-1)^2 + x(2)^2;
end

%% 快速非支配排序
function F = fast_non_domination_sort(pop)
    n = length(pop);
    S = cell(n,1);
    F = struct('ids',[],'n',[]);
    F(1).ids = [];
    F(1).n = zeros(n,1);
    for p = 1 : n
        S{p} = [];
        for q = 1 : n
            if dominates(pop(p).f, pop(q).f)
                S{p} = [S{p} q];
            elseif dominates(pop(q).f, pop(p).f)
                pop(p).n = pop(p).n + 1;
            end
        end
        if pop(p).n == 0
            F(1).ids = [F(1).ids p];
        end
    end
    i = 1;
    while ~isempty(F(i).ids)
        Q = [];
        for p = F(i).ids
            for q = S{p}
                pop(q).n = pop(q).n - 1;
                if pop(q).n == 0
                    Q = [Q q];
                end
            end
        end
        i = i + 1;
        F(i).ids = Q;
    end
    i = 1;
    for f = 1 : length(F)
        for p = F(f).ids
            pop(p).rank = i;
        end
        i = i + 1;
    end
end

%% 计算拥挤度距离
function pop = calculate_crowding_distance(pop)
    n = length(pop);
    for i = 1 : n
        pop(i).dist = 0;
    end
    for m = 1 : 2
        [~,idx] = sort([pop.f(:,m)]);
        pop(idx(1)).dist = Inf;
        pop(idx(n)).dist = Inf;
        for i = 2 : n-1
            pop(idx(i)).dist = pop(idx(i)).dist + (pop(idx(i+1)).f(m) - pop(idx(i-1)).f(m)) / (max([pop.f(:,m)]) - min([pop.f(:,m)]));
        end
    end
end

%% 选择
function pop = selection(pop, F)
    pop_size = length(pop);
    n = length(F);
    cum_size = zeros(n,1);
    for i = 1 : n
        cum_size(i) = length(F(i).ids);
    end
    cum_size = cumsum(cum_size);
    new_pop = repmat(struct('x',[],'f',[],'rank',[],'dist',[]), pop_size, 1);
    for i = 1 : pop_size
        if i <= cum_size(1)
            f_idx = 1;
        else
            f_idx = find(cum_size>=i,1);
        end
        p_idx = F(f_idx).ids(randi(length(F(f_idx).ids)));
        new_pop(i) = pop(p_idx);
    end
    pop = new_pop;
end

%% 交叉
function pop = crossover(pop, pc, eta_c)
    pop_size = length(pop);
    offspring = repmat(struct('x',[],'f',[],'rank',[],'dist',[]), pop_size, 1);
    for i = 1 : 2 : pop_size
        if rand() < pc
            p1 = pop(randi(pop_size));
            p2 = pop(randi(pop_size));
            beta = rand(1,2) .* (1+2*eta_c) - eta_c;
            beta(beta<0) = 0;
            beta(beta>1) = 1;
            c1 = beta(1)*p1.x + (1-beta(1))*p2.x;
            c2 = beta(2)*p2.x + (1-beta(2))*p1.x;
            c1 = bound_check(c1,l_bound,u_bound);
            c2 = bound_check(c2,l_bound,u_bound);
            offspring(i).x = c1;
            offspring(i+1).x = c2;
        else
            offspring(i) = pop(i);
            offspring(i+1) = pop(i+1);
        end
    end
    pop = offspring;
end

%% 变异
function pop = mutation(pop, pm, eta_m, l_bound, u_bound)
    pop_size = length(pop);
    for i = 1 : pop_size
        if rand() < pm
            p = pop(i);
            beta = rand(size(p.x));
            beta(beta<0.5) = (2*beta(beta<0.5)).^(1/(eta_m+1));
            beta(beta>=0.5) = (2-2*beta(beta>=0.5)).^(-1/(eta_m+1));
            c = p.x + (u_bound-l_bound).*beta.*(rand(size(p.x))-0.5);
            c = bound_check(c,l_bound,u_bound);
            pop(i).x = c;
        end
    end
end

%% 越界检查
function x = bound_check(x, l_bound, u_bound)
    x(x<l_bound) = l_bound(x<l_bound);
    x(x>u_bound) = u_bound(x>u_bound);
end

%% 支配关系
function flag = dominates(f1, f2)
    flag = all(f1 <= f2) && any(f1 < f2);
end

%% 淘汰
function pop = replace(new_pop, pop_size)
    pop = repmat(struct('x',[],'f',[],'rank',[],'dist',[]), pop_size, 1);
    F = fast_non_domination_sort(new_pop);
    n = 0;
    i = 1;
    while n + length(F(i).ids) <= pop_size
        for j = 1 : length(F(i).ids)
            pop(n+j) = new_pop(F(i).ids(j));
        end
        n = n + length(F(i).ids);
        i = i + 1;
    end
    if n < pop_size
        [~,idx] = sort([new_pop(F(i).ids).dist],'descend');
        for j = 1 : pop_size - n
            pop(n+j) = new_pop(F(i).ids(idx(j)));
        end
    end
end