CCMO是采用双种群协作的方式，使用NSGAII框架；另外，为了提高多样性，替代NSGAII的拥挤度距离(选择最不拥挤)，采用SPEA2的裁剪策略(删除最拥挤的)。

一、CCMO算法流程

代码见附录 CCMO.md

二、为什么CCMO是弱协同

三、NSGAII和SPEA2对比

SPEA2的环境选择机制

1、适应度计算：

• • 强度值计算（Strength Value）：计算每个解支配的其他解的数量。
  • 适应度值计算（Fitness Value）：计算每个解的适应度值，适应度值等于所有被当前解支配的解的强度值之和，再加上一个微小的常数来处理未被任何解支配的情况。

2、选择非支配解：

• • 选择适应度小于1的解，这些解通常是非支配解。

3、裁剪多余解：

• • 如果选择的非支配解数量超过种群大小，通过截断算法删除多余的解。截断算法根据解的拥挤度（即距离）来选择删除哪些解，以保持种群的多样性。

NSGA-II的非支配排序机制

1、非支配排序：

• • 将种群中的个体根据支配关系分成不同的等级（Front）。第一个等级（Front 1）包含所有非支配解，第二个等级（Front 2）包含被Front 1支配的解，以此类推。

2、拥挤距离计算：

• • 对每个前沿（Front）的个体计算拥挤距离，拥挤距离用于衡量个体在目标空间的稀疏程度。

3、环境选择：

• • 优先选择低等级（高优先级）的前沿个体。如果某个前沿的个体数量超过剩余的种群大小，根据拥挤距离选择个体，以保持种群的多样性。

四、附录

CCMO.m

classdef CCMO < ALGORITHM
% <multi> <real/integer/label/binary/permutation> <constrained>
% Coevolutionary constrained multi-objective optimization framework
% type --- 1 --- Type of operator (1. GA 2. DE)

%------------------------------- Reference --------------------------------
% Y. Tian, T. Zhang, J. Xiao, X. Zhang, and Y. Jin, A coevolutionary
% framework for constrained multi-objective optimization problems, IEEE
% Transactions on Evolutionary Computation, 2021, 25(1): 102-116.
%------------------------------- Copyright --------------------------------
% Copyright (c) 2024 BIMK Group. You are free to use the PlatEMO for
% research purposes. All publications which use this platform or any code
% in the platform should acknowledge the use of "PlatEMO" and reference "Ye
% Tian, Ran Cheng, Xingyi Zhang, and Yaochu Jin, PlatEMO: A MATLAB platform
% for evolutionary multi-objective optimization [educational forum], IEEE
% Computational Intelligence Magazine, 2017, 12(4): 73-87".
%--------------------------------------------------------------------------

    methods
        function main(Algorithm,Problem)
            %% Parameter setting
            type = Algorithm.ParameterSet(1); % 默认采用GA算法

            %% 初始化两个种群(解和目标值)
            Population1 = Problem.Initialization(); % 主种群，带约束
            Population2 = Problem.Initialization(); % 辅助种群，无约束

            %% 分别评价两个种群，求适应度(被支配的数量)
            Fitness1    = CalFitness(Population1.objs,Population1.cons);
            Fitness2    = CalFitness(Population2.objs);

            %% Optimization
            while Algorithm.NotTerminated(Population1)
                if type == 1 % 采用GA算法
                    %% 分别选择父代(二元锦标赛：从N个个体里挑选2个个体，比较适应度，选择好的那个。重复N次，可重复选择)
                    MatingPool1 = TournamentSelection(2,Problem.N,Fitness1);
                    MatingPool2 = TournamentSelection(2,Problem.N,Fitness2);

                    %% 分别生成子代(采用GA算法，对上面的种群进行模拟二进制交叉、二项式变异)
                    Offspring1  = OperatorGAhalf(Problem,Population1(MatingPool1));
                    Offspring2  = OperatorGAhalf(Problem,Population2(MatingPool2));

                elseif type == 2
                    MatingPool1 = TournamentSelection(2,2*Problem.N,Fitness1);
                    MatingPool2 = TournamentSelection(2,2*Problem.N,Fitness2);
                    Offspring1  = OperatorDE(Problem,Population1,Population1(MatingPool1(1:end/2)),Population1(MatingPool1(end/2+1:end)));
                    Offspring2  = OperatorDE(Problem,Population2,Population2(MatingPool2(1:end/2)),Population2(MatingPool2(end/2+1:end)));
                end

                %% 选择更新(优先选择非支配解；如果非支配太多了，采用SPEA2的截断策略，循环删除最拥挤的个体)
                [Population1,Fitness1] = EnvironmentalSelection([Population1,Offspring1,Offspring2],Problem.N,true);
                [Population2,Fitness2] = EnvironmentalSelection([Population2,Offspring1,Offspring2],Problem.N,false);
            end
        end
    end
end

CalFitness.m

function Fitness = CalFitness(PopObj,PopCon)
% Calculate the fitness of each solution
% 求适应度，这里不是求目标值，而是求每个解被支配的数量

%------------------------------- Copyright --------------------------------
% Copyright (c) 2024 BIMK Group. You are free to use the PlatEMO for
% research purposes. All publications which use this platform or any code
% in the platform should acknowledge the use of "PlatEMO" and reference "Ye
% Tian, Ran Cheng, Xingyi Zhang, and Yaochu Jin, PlatEMO: A MATLAB platform
% for evolutionary multi-objective optimization [educational forum], IEEE
% Computational Intelligence Magazine, 2017, 12(4): 73-87".
%--------------------------------------------------------------------------

    N = size(PopObj,1);
    if nargin == 1 % 如果只有一个输入参数
        CV = zeros(N,1);
    else
        CV = sum(max(0,PopCon),2); % 计算约束违约度CV，每个个体的约束违约度为其所有约束违约量的和
    end

    %% Detect the dominance relation between each two solutions
    Dominate = false(N); % 初始化支配关系矩阵，大小为N x N，全为false
    for i = 1 : N-1
        for j = i+1 : N
            if CV(i) < CV(j)
                Dominate(i,j) = true; % 如果个体i的约束违约度小于个体j，则个体i支配个体j
            elseif CV(i) > CV(j)
                Dominate(j,i) = true; % 如果个体i的约束违约度大于个体j，则个体j支配个体i
            else
                k = any(PopObj(i,:)<PopObj(j,:)) - any(PopObj(i,:)>PopObj(j,:));
                if k == 1
                    Dominate(i,j) = true; % i个体至少一个目标优于j，并且没有任何目标劣于j
                elseif k == -1
                    Dominate(j,i) = true; % j个体至少一个目标优于i，并且没有任何目标劣于i
                end
            end
        end
    end
    
    %% 每个解支配其他解的数量
    S = sum(Dominate,2); % 2表示行，即求所有行之和，返回一个列向量，代表每个个体直接别人的数量
    
    %% 每个解被多少个解支配
    R = zeros(1,N); 
    for i = 1 : N
        R(i) = sum(S(Dominate(:,i)));
    end
    
    %% Calculate D(i) 每个解的拥挤度距离
    Distance = pdist2(PopObj,PopObj);  % 计算种群中每两个个体之间的距离矩阵
    Distance(logical(eye(length(Distance)))) = inf; % 将距离矩阵的对角线元素设为无穷大，以避免自身比较
    Distance = sort(Distance,2);  % 对距离矩阵的每一行进行排序
    D = 1./(Distance(:,floor(sqrt(N)))+2); % 计算拥挤距离 D，Nx1
    
    %% Calculate the fitnesses 适应度为被支配程度和拥挤距离之和
    % R是1xN，D是Nx1，D'是1xN
    Fitness = R + D';  % 这里的 D' 是 D 的转置操作，将列向量 D 转换为行向量，以便与 R 进行逐元素相加
end

function [Population,Fitness] = EnvironmentalSelection(Population,N,isOrigin)
% The environmental selection of SPEA2
%%  SPEA2的选择和NSGAII的选择显著不同

%------------------------------- Copyright --------------------------------
% Copyright (c) 2024 BIMK Group. You are free to use the PlatEMO for
% research purposes. All publications which use this platform or any code
% in the platform should acknowledge the use of "PlatEMO" and reference "Ye
% Tian, Ran Cheng, Xingyi Zhang, and Yaochu Jin, PlatEMO: A MATLAB platform
% for evolutionary multi-objective optimization [educational forum], IEEE
% Computational Intelligence Magazine, 2017, 12(4): 73-87".
%--------------------------------------------------------------------------

    % Population = [Population1,Offspring1,Offspring2]，故大小大于N

    %% Calculate the fitness of each solution
    if isOrigin
        Fitness = CalFitness(Population.objs,Population.cons);
    else
        Fitness = CalFitness(Population.objs);
    end

    %% Environmental selection
    % 选择适应度小于1的解（这些解通常是非支配解）。
    Next = Fitness < 1;
    if sum(Next) < N % 如果选择的解数量小于所需的种群大小 N
        [~,Rank] = sort(Fitness); % 按适应度排序，选择适应度最好的前 N 个解
        Next(Rank(1:N)) = true;
    elseif sum(Next) > N % 如果选择的解数量超过所需的种群大小 N
        Del  = Truncation(Population(Next).objs,sum(Next)-N); % 调用 Truncation 函数，通过截断删除多余的解
        Temp = find(Next);
        Next(Temp(Del)) = false;
    end
    % Population for next generation
    Population = Population(Next);
    Fitness    = Fitness(Next);
    % Sort the population
    [Fitness,rank] = sort(Fitness);
    Population = Population(rank);
end

%% 根据拥挤度删除多余的最拥挤的个体，直到删除K个
function Del = Truncation(PopObj,K)

    %% 裁剪，
    % SPEA2是直接循环删除最拥挤的；
    % NSGAII是先选前沿，前沿在某层超了，则保留那层两边的，然后选择拥挤度小的
    Distance = pdist2(PopObj,PopObj);  % 计算种群中每两个个体之间的距离矩阵
    Distance(logical(eye(length(Distance)))) = inf; % 将距离矩阵的对角线元素设为无穷大，以避免自身比较
    Del = false(1,size(PopObj,1)); % 初始化删除标记
    while sum(Del) < K
        Remain   = find(~Del); % 找到未被删除的个体
        Temp     = sort(Distance(Remain,Remain),2);  % 对距离进行排序
        [~,Rank] = sortrows(Temp); % 对排序结果进行排序，得到最拥挤的个体
        Del(Remain(Rank(1))) = true; % 删除最拥挤的个体
    end
end