多目标优化拥挤距离计算

拥挤距离主要是维持种群中个体的多样性。具体而言，一般来说是指种群按照支配关系[1]进行非支配排序[2]后,单个 Rank 层中个体的密集程度。常用于支配关系的多目标算法中，例如NSGA-II[3].
主要步骤如下：
1. 取单个前沿中个体按照一个目标上的值从小到大排序
2. 将最大目标值作为 max,最小目标值保留作为 min。并且这两个极值点的拥挤距离都被设置为 inf 即无穷大。因此注意，一个层中可能有多个具有 inf 的点，即如果层中有多个点在至少一个目标上相等，并且最大或最小，那么这些点的拥挤距离都是无穷大！！因为目标上呈现垂直的关系也是属于非支配的关系！！如果出现这种情况，说明你算法的多样性很烂！~或者在某些算法早期可能出现这种情况
3. 在这个目标上计算每个个体最相邻个体之间的距离，即 i-1 和 i+1 的目标值的差。并使用 max 和 min 对次值进行归一化。
4. 遍历目标，将目标上已经归一化的拥挤距离相加。
5. 进入下一层 front 前沿
6. 拥挤距离越大越好，最后按照拥挤距离重新排序各层，进而排序种群

matlab

function CrowdDis = CrowdingDistance(PopObj)
% Calculate the crowding distance of each solution in the same front

    [N,M]    = size(PopObj);

    CrowdDis = zeros(1,N);
    Fmax     = max(PopObj,[],1);
    Fmin     = min(PopObj,[],1);
    for i = 1 : M
        [~,rank] = sortrows(PopObj(:,i));
        CrowdDis(rank(1))   = inf;
        CrowdDis(rank(end)) = inf;
        for j = 2 : N-1
            CrowdDis(rank(j)) = CrowdDis(rank(j))+(PopObj(rank(j+1),i)-PopObj(rank(j-1),i))/(Fmax(i)-Fmin(i));
        end
    end
end

jmetal

public void crowdingDistanceAssignment(SolutionSet solutionSet, int nObjs) {
        int size = solutionSet.size();

        if (size == 0)
            return;

        if (size == 1) {
            solutionSet.get(0).setCrowdingDistance(Double.POSITIVE_INFINITY);
            return;
        } // if

        if (size == 2) {
            solutionSet.get(0).setCrowdingDistance(Double.POSITIVE_INFINITY);
            solutionSet.get(1).setCrowdingDistance(Double.POSITIVE_INFINITY);
            return;
        } // if

        // Use a new SolutionSet to evite alter original solutionSet
        SolutionSet front = new SolutionSet(size);
        for (int i = 0; i < size; i++) {
            front.add(solutionSet.get(i));
        }

        for (int i = 0; i < size; i++)
            front.get(i).setCrowdingDistance(0.0);

        double objetiveMaxn;
        double objetiveMinn;
        double distance;

        for (int i = 0; i < nObjs; i++) {
            // Sort the population by Obj n
            front.sort(new ObjectiveComparator(i));
            objetiveMinn = front.get(0).getObjective(i);
            objetiveMaxn = front.get(front.size() - 1).getObjective(i);

            // Set de crowding distance
            front.get(0).setCrowdingDistance(Double.POSITIVE_INFINITY);
            front.get(size - 1).setCrowdingDistance(Double.POSITIVE_INFINITY);

            for (int j = 1; j < size - 1; j++) {
                distance = front.get(j + 1).getObjective(i) - front.get(j - 1).getObjective(i);
                distance = distance / (objetiveMaxn - objetiveMinn);
                distance += front.get(j).getCrowdingDistance();
                front.get(j).setCrowdingDistance(distance);
            } // for
        } // for
    } // crowdingDistanceAssing

参考资料

[1]支配关系: https://blog.csdn.net/u013555719/article/details/91356078

[2]非支配排序: https://blog.csdn.net/u013555719/article/details/105564693

[3]NSGA-II: https://blog.csdn.net/u013555719/article/details/82936554

​多目标优化拥挤距离计算

多目标优化拥挤距离计算

matlab

jmetal

参考资料

多目标优化拥挤距离计算