SBX算法和PM算法】

SBX算法和PM算法都是进化算法中常用的优化算法，用于解决优化问题。下面分别介绍一下这两种算法：

SBX算法（Simulated Binary Crossover）是一种模拟二进制交叉算法。它主要用于遗传算法中的交叉操作，用于产生新的个体。SBX算法通过模拟二进制交叉，将两个个体的基因进行交换和调整，生成两个新的个体。在交叉过程中，通过调整交叉概率和分布指数来控制交叉的强度和多样性。SBX算法的特点是能够有效地保持父代个体的特征，并引入新的个体变异，提高了全局搜索能力。

PM算法（Polynomial Mutation）是一种多项式变异算法，主要用于遗传算法中的变异操作。PM算法通过多项式分布来对个体的基因进行变异，引入随机扰动，从而增加个体的多样性。在变异过程中，可以通过调整变异概率和分布指数来控制变异的强度和多样性。PM算法的特点是能够快速引入新的个体变异，并且具有一定的局部搜索能力。

总结来说，SBX算法主要用于交叉操作，通过模拟二进制交叉来产生新的个体，增加全局搜索能力。而PM算法主要用于变异操作，通过多项式变异来引入随机扰动，增加个体的多样性，具有一定的局部搜索能力。这两种算法在进化算法中经常被应用于优化问题的求解。

相关问题

nsga-II算法的matlab代码

以下是NSGA-II算法的MATLAB代码：

%% NSGA-II Algorithm
function [pop, front] = NSGAII(pop_size,gen_size,vari_num,x_min,x_max)

% Parameters Setting
pc = 0.9;   % crossover probability
pm = 1/vari_num; % mutation probability
eta_c = 20; % crossover distribution index
eta_m = 20; % mutation distribution index
pop = rand(pop_size,vari_num) .* repmat((x_max-x_min),pop_size,1) + repmat(x_min,pop_size,1); % initial population
for i = 1:pop_size
    obj(i,1) = f1(pop(i,:));
    obj(i,2) = f2(pop(i,:));
end

% NSGA-II Algorithm
for i = 1:gen_size
    % Non-dominated Sorting
    [front,~] = non_domination_sort(obj);
    % Crowding Distance Calculation
    for j = 1:length(front)
        crowd_dis(j,:) = crowding_distance(obj(front{j},:));
    end
    % Mating Selection & Variation
    pop_new = zeros(pop_size,vari_num);
    obj_new = zeros(pop_size,2);
    count = 0;
    for j = 1:length(front)
        [temp,index] = sort(crowd_dis(j,:),'descend');
        front_member = front{j};
        pop_temp = pop(front_member,:);
        for k = 1:length(front_member)
            if rand < pc && k ~= length(front_member)
                p1 = front_member(k);
                p2 = front_member(k+1);
                pop_new(count+1,:) = crossover(pop_temp(p1,:),pop_temp(p2,:),eta_c);
                obj_new(count+1,:) = [f1(pop_new(count+1,:)),f2(pop_new(count+1,:))];
                count = count + 1;
            elseif rand < pm
                p = front_member(k);
                pop_new(count+1,:) = mutation(pop_temp(p,:),eta_m,x_min,x_max);
                obj_new(count+1,:) = [f1(pop_new(count+1,:)),f2(pop_new(count+1,:))];
                count = count + 1;
            end
        end
        if count >= pop_size
            break;
        end
    end
    % Combine Parent & Offspring Populations
    pop = [pop;pop_new(1:pop_size-count,:)];
    obj = [obj;obj_new(1:pop_size-count,:)];
end

% Results Output
[front,~] = non_domination_sort(obj);
for i = 1:length(front)
    plot(obj(front{i},1),obj(front{i},2),'o');
    hold on;
end
xlabel('f_1');
ylabel('f_2');
end

%% Non-dominated Sorting
function [front,rank] = non_domination_sort(obj)

n = size(obj,1);
rank = inf(1,n);
dominate = false(n);

for i = 1:n
    for j = i+1:n
        if all(obj(i,:) <= obj(j,:)) && any(obj(i,:) < obj(j,:))
            dominate(i,j) = true;
        elseif all(obj(i,:) >= obj(j,:)) && any(obj(i,:) > obj(j,:))
            dominate(j,i) = true;
        end
    end
end

for i = 1:n
    if sum(dominate(:,i)) == 0
        rank(i) = 1;
    end
end

front{1} = find(rank == 1);

i = 1;
while ~isempty(front{i})
    Q = [];
    for j = front{i}
        for k = 1:n
            if dominate(j,k)
                dominate(j,k) = false;
                if sum(dominate(k,:)) == 0
                    rank(k) = i + 1;
                    Q = [Q,k];
                end
            end
        end
    end
    i = i + 1;
    front{i} = Q;
end
end

%% Crowding Distance Calculation
function [crowd_dis] = crowding_distance(obj)

n = size(obj,1);
crowd_dis = zeros(1,n);
f_max = max(obj,[],1);
f_min = min(obj,[],1);

for i = 1:size(obj,2)
    [~,index] = sort(obj(:,i));
    crowd_dis(index(1)) = inf;
    crowd_dis(index(end)) = inf;
    for j = 2:n-1
        crowd_dis(index(j)) = crowd_dis(index(j)) + (obj(index(j+1),i) - obj(index(j-1),i))/(f_max(i) - f_min(i));
    end
end
end

%% SBX Crossover Operator
function [offspring] = crossover(p1,p2,eta_c)

n = length(p1);
u = rand(1,n);
offspring = zeros(1,n);

for i = 1:n
    if u(i) <= 0.5
        if abs(p1(i)-p2(i)) > 1e-10
            if p1(i) < p2(i)
                x1 = p1(i);
                x2 = p2(i);
            else
                x1 = p2(i);
                x2 = p1(i);
            end
            y1 = (x1 - floor(x1)) + floor(x2);
            y2 = (x2 - floor(x2)) + floor(x1);
            if rand < 0.5
                offspring(i) = y1;
            else
                offspring(i) = y2;
            end
        else
            offspring(i) = p1(i);
        end
    else
        if abs(p1(i)-p2(i)) > 1e-10
            if p1(i) < p2(i)
                x1 = p1(i);
                x2 = p2(i);
            else
                x1 = p2(i);
                x2 = p1(i);
            end
            y1 = 2*x1 - x2;
            y2 = 2*x2 - x1;
            if y1 < 0
                y1 = 0;
            elseif y1 > 1
                y1 = 1;
            end
            if y2 < 0
                y2 = 0;
            elseif y2 > 1
                y2 = 1;
            end
            offspring(i) = y1;
        else
            offspring(i) = p1(i);
        end
    end
end
end

%% Polynomial Mutation Operator
function [offspring] = mutation(p,eta_m,x_min,x_max)

n = length(p);
offspring = p;

for i = 1:n
    if rand < eta_m/n
        delta1 = (p(i) - x_min)/(x_max - x_min);
        delta2 = (x_max - p(i))/(x_max - x_min);
        u = rand;
        if u <= 0.5
            deltaq = (2*u + (1 - 2*u)*(1-delta1)^(eta_m+1))^(1/(eta_m+1))-1;
        else
            deltaq = 1 - (2*(1-u) + 2*(u-0.5)*(1-delta2)^(eta_m+1))^(1/(eta_m+1));
        end
        offspring(i) = p(i) + deltaq*(x_max - x_min);
        if offspring(i) < x_min
            offspring(i) = x_min;
        elseif offspring(i) > x_max
            offspring(i) = x_max;
        end
    end
end
end

%% Objective Function 1
function [y1] = f1(x)

y1 = x(1);

end

%% Objective Function 2
function [y2] = f2(x)

y2 = (1+x(2))/x(1);

end

其中，`pop_size`为种群大小，`gen_size`为迭代次数，`vari_num`为变量个数，`x_min`和`x_max`分别为变量的上下界。`f1`和`f2`分别为两个目标函数，根据具体问题进行修改。

如何通过python实现nsga-Ⅱ算法

在Python中实现NSGA-II算法，可以使用一些开源的多目标优化库，例如pymoo、deap等。下面以pymoo库为例，介绍如何实现NSGA-II算法。

首先，我们需要定义优化问题的目标函数和约束条件（如果有的话）。这里以一个简单的二元优化问题为例：

import numpy as np

def obj_func(x):
    f1 = x[0]**2 + x[1]**2
    f2 = (x[0]-1)**2 + x[1]**2
    return [f1, f2]

# 定义变量的上下限
xl, xu = np.array([-5, -5]), np.array([5, 5])

接下来，我们可以使用pymoo库中的NSGA-II算法来解决该问题。具体代码如下：

from pymoo.algorithms.nsga2 import NSGA2
from pymoo.factory import get_crossover, get_mutation, get_sampling
from pymoo.model.problem import Problem
from pymoo.optimize import minimize

class MyProblem(Problem):
    def __init__(self):
        super().__init__(n_var=2,
                         n_obj=2,
                         n_constr=0,
                         xl=xl,
                         xu=xu)

    def _evaluate(self, x, out, *args, **kwargs):
        out["F"] = obj_func(x)

# 定义NSGA-II算法的参数
algorithm = NSGA2(pop_size=100,
                  n_offsprings=50,
                  sampling=get_sampling("real_random"),
                  crossover=get_crossover("real_sbx", prob=0.9, eta=15),
                  mutation=get_mutation("real_pm", prob=0.1, eta=20),
                  eliminate_duplicates=True)

# 定义优化问题
problem = MyProblem()

# 运行NSGA-II算法，得到最优解
res = minimize(problem,
               algorithm,
               ('n_gen', 100),
               verbose=True)

以上代码中，我们首先定义了一个继承自pymoo库中Problem类的自定义问题类MyProblem，其中重写了_evaluate方法来计算目标函数值。然后，我们使用NSGA2类来创建一个NSGA-II算法实例，并设置一些参数，例如种群大小、交叉和变异算子等。最后，我们将问题和算法实例传入minimize函数中，得到最优解。

需要注意的是，pymoo库还提供了许多其他优化算法和问题，具体使用方法可以参考官方文档。