NSGA-II非支配排序遗传算法

NSGA-II（非支配排序遗传算法II）

NSGA-II是一种多目标优化算法，广泛应用于解决具有多个目标的优化问题。与传统的遗传算法相比，NSGA-II通过非支配排序和拥挤距离机制，能够有效避免陷入局部最优，属于全局优化算法。它在多目标优化领域表现出色，能够同时优化多个目标，并生成一组非支配解（Pareto前沿）。

1. NSGA-II的基本原理

1.1 非支配排序

非支配排序是一种用于评估个体在多目标优化中的优势的方法。如果一个解在所有目标上都不比另一个解差，并且至少在一个目标上比另一个解好，则称这个解非支配另一个解。通过非支配排序，可以将种群分为多个非支配层。

1.2 拥挤距离

拥挤距离用于评估个体在非支配层中的分布情况。通过计算每个个体在目标空间中的平均距离，可以避免解的过度集中，从而保持种群的多样性。

1.3 遗传操作

NSGA-II结合了选择、交叉和变异等遗传操作，通过这些操作生成新的种群，并逐步逼近最优解。

2. MATLAB实现

2.1 初始化种群

function population = initializePopulation(populationSize, numVariables, lowerBound, upperBound)
    population = zeros(populationSize, numVariables);
    for i = 1:populationSize
        population(i, :) = lowerBound + (upperBound - lowerBound) .* rand(1, numVariables);
    end
end

2.2 非支配排序

function [fronts, crowdingDistance] = nonDominatedSorting(population, objectives)
    populationSize = size(population, 1);
    fronts = {};
    front = [];
    dominatedCount = zeros(populationSize, 1);
    dominates = cell(populationSize, 1);
    crowdingDistance = zeros(populationSize, 1);

    for i = 1:populationSize
        for j = 1:populationSize
            if i ~= j
                if all(objectives(i, :) <= objectives(j, :)) && any(objectives(i, :) < objectives(j, :))
                    dominates{i} = [dominates{i}; j];
                elseif all(objectives(j, :) <= objectives(i, :)) && any(objectives(j, :) < objectives(i, :))
                    dominatedCount(i) = dominatedCount(i) + 1;
                end
            end
        end
    end

    frontIndex = find(dominatedCount == 0);
    front = frontIndex;
    fronts{1} = front;

    while ~isempty(front)
        nextFront = [];
        for i = front
            for j = dominates{i}
                dominatedCount(j) = dominatedCount(j) - 1;
                if dominatedCount(j) == 0
                    nextFront = [nextFront; j];
                end
            end
        end
        front = nextFront;
        if ~isempty(front)
            fronts{end+1} = front;
        end
    end

    for i = 1:length(fronts)
        front = fronts{i};
        for j = 1:size(objectives, 2)
            [sortedObjectives, sortIndex] = sort(objectives(front, j));
            crowdingDistance(sortIndex) = crowdingDistance(sortIndex) + [0; diff(sortedObjectives)];
        end
    end
end

2.3 拥挤距离排序

function sortedIndex = crowdingDistanceSorting(crowdingDistance)
    [~, sortedIndex] = sort(crowdingDistance, 'descend');
end

2.4 选择操作

function selectedPopulation = selection(population, objectives, populationSize)
    [fronts, crowdingDistance] = nonDominatedSorting(population, objectives);
    selectedPopulation = [];
    for i = 1:length(fronts)
        front = fronts{i};
        if size(selectedPopulation, 1) + length(front) <= populationSize
            selectedPopulation = [selectedPopulation; population(front, :)];
        else
            remaining = populationSize - size(selectedPopulation, 1);
            sortedIndex = crowdingDistanceSorting(crowdingDistance(front));
            selectedPopulation = [selectedPopulation; population(front(sortedIndex(1:remaining)), :)];
            break;
        end
    end
end

2.5 交叉操作

function offspring = crossover(parent1, parent2, crossoverRate)
    if rand < crossoverRate
        crossoverPoint = randi([1, size(parent1, 2)]);
        offspring1 = [parent1(1:crossoverPoint), parent2(crossoverPoint+1:end)];
        offspring2 = [parent2(1:crossoverPoint), parent1(crossoverPoint+1:end)];
    else
        offspring1 = parent1;
        offspring2 = parent2;
    end
    offspring = [offspring1; offspring2];
end

2.6 变异操作

function mutatedIndividual = mutation(individual, mutationRate, lowerBound, upperBound)
    for i = 1:length(individual)
        if rand < mutationRate
            individual(i) = lowerBound(i) + (upperBound(i) - lowerBound(i)) * rand;
        end
    end
    mutatedIndividual = individual;
end

2.7 主函数

function [population, objectives] = NSGA2(populationSize, numVariables, numObjectives, lowerBound, upperBound, maxGenerations, crossoverRate, mutationRate)
    % 初始化种群
    population = initializePopulation(populationSize, numVariables, lowerBound, upperBound);

    % 评估初始种群
    objectives = evaluateObjectives(population);

    for generation = 1:maxGenerations
        % 选择操作
        selectedPopulation = selection(population, objectives, populationSize);

        % 交叉操作
        offspring = [];
        for i = 1:2:populationSize
            parent1 = selectedPopulation(i, :);
            parent2 = selectedPopulation(i+1, :);
            offspring = [offspring; crossover(parent1, parent2, crossoverRate)];
        end

        % 变异操作
        for i = 1:size(offspring, 1)
            offspring(i, :) = mutation(offspring(i, :), mutationRate, lowerBound, upperBound);
        end

        % 合并种群
        combinedPopulation = [population; offspring];
        combinedObjectives = [objectives; evaluateObjectives(offspring)];

        % 选择下一代
        [population, objectives] = selection(combinedPopulation, combinedObjectives, populationSize);

        % 显示进度
        fprintf('Generation %d\n', generation);
    end
end

function objectives = evaluateObjectives(population)
    % 示例目标函数
    objectives = zeros(size(population, 1), 2);
    for i = 1:size(population, 1)
        objectives(i, 1) = sum(population(i, :).^2);
        objectives(i, 2) = sum((population(i, :) - 1).^2);
    end
end

3. 调用NSGA-II

% 参数设置
populationSize = 100;
numVariables = 2;
numObjectives = 2;
lowerBound = [0, 0];
upperBound = [1, 1];
maxGenerations = 100;
crossoverRate = 0.9;
mutationRate = 0.1;

% 调用NSGA-II
[population, objectives] = NSGA2(populationSize, numVariables, numObjectives, lowerBound, upperBound, maxGenerations, crossoverRate, mutationRate);

% 可视化结果
figure;
plot(objectives(:, 1), objectives(:, 2), 'o');
xlabel('Objective 1');
ylabel('Objective 2');
title('Pareto Front');

参考代码 NSGA-II非支配排序遗传算法，可避免陷入局部最优，属于全局算法 youwenfan.com/contentcnb/79674.html

4. 结论

NSGA-II是一种强大的多目标优化算法，能够有效避免陷入局部最优，属于全局优化算法。通过非支配排序和拥挤距离机制，NSGA-II能够生成一组非支配解，适用于解决复杂的多目标优化问题。