基于自适应差分进化算法的MATLAB实现

基于自适应差分进化算法（Adaptive Differential Evolution, ADE）的MATLAB实现

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一、算法原理

自适应差分进化算法通过动态调整缩放因子（F）和交叉概率（CR）提升全局搜索能力。核心改进包括：

1. 参数自适应：根据种群适应度动态调整F和CR
2. 变异策略：采用DE/rand/1策略生成变异向量
3. 精英保留：始终保留当前最优解

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二、代码

function [best_sol, best_fit, history] = AdaptiveDE(func, lb, ub, dim, NP, max_iter)
    % 参数初始化
    F_min = 0.1;  F_max = 0.9;  % 缩放因子范围
    CR_min = 0.1; CR_max = 0.9; % 交叉概率范围
    tau1 = 0.1; tau2 = 0.1;     % 参数更新概率
    
    % 种群初始化
    pop = repmat(lb, NP, 1) + rand(NP, dim) .* repmat(ub-lb, NP, 1);
    fitness = arrayfun(@(i) func(pop(i,:)), 1:NP);
    
    % 记录最优解
    [best_fit, best_idx] = min(fitness);
    best_sol = pop(best_idx, :);
    history = zeros(max_iter, 1);
    
    % 主循环
    for gen = 1:max_iter
        % 自适应参数更新
        F = F_min + (F_max-F_min) * rand(NP,1);
        CR = CR_min + (CR_max-CR_min) * rand(NP,1);
        
        for i = 1:NP
            % 变异操作
            r1 = randi([1,NP]); while r1==i, r1=randi([1,NP]); end
            r2 = randi([1,NP]); while r2==i || r2==r1, r2=randi([1,NP]); end
            r3 = randi([1,NP]); while r3==i || r3==r1 || r3==r2, r3=randi([1,NP]); end
            V = pop(r1,:) + F(i) * (pop(r2,:) - pop(r3,:));
            
            % 交叉操作
            j_rand = randi(dim);
            U = pop(i,:);
            for j = 1:dim
                if rand() < CR(i) || j == j_rand
                    U(j) = V(j);
                end
            end
            
            % 边界处理
            U = max(U, lb);
            U = min(U, ub);
            
            % 选择操作
            new_fit = func(U);
            if new_fit < fitness(i)
                pop(i,:) = U;
                fitness(i) = new_fit;
                
                % 参数更新
                if rand() < tau1
                    F(i) = F_min + rand()*(F_max-F_min);
                end
                if rand() < tau2
                    CR(i) = CR_min + rand()*(CR_max-CR_min);
                end
            end
        end
        
        % 更新最优解
        [current_best, idx] = min(fitness);
        if current_best < best_fit
            best_fit = current_best;
            best_sol = pop(idx, :);
        end
        history(gen) = best_fit;
        
        % 显示进度
        fprintf('Generation %d: Best Fitness = %.6f\n', gen, best_fit);
    end
end

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三、关键代码解析

1. 参数自适应机制

% 自适应参数更新
F = F_min + (F_max-F_min) * rand(NP,1);
CR = CR_min + (CR_max-CR_min) * rand(NP,1);

• 每个个体独立调整F和CR
• 参数范围动态调整（[0.1,0.9]）

2. 变异与交叉操作

% DE/rand/1变异策略
V = pop(r1,:) + F(i) * (pop(r2,:) - pop(r3,:));

% 二项交叉操作
for j = 1:dim
    if rand() < CR(i) || j == j_rand
        U(j) = V(j);
    end
end

• 保证至少一个维度来自变异向量
• 随机维度确保多样性

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四、测试函数示例

1. Rastrigin函数测试

% 定义目标函数
rastrigin = @(x) 10*numel(x) + sum(x.^2 - 10*cos(2*pi*x));

% 参数设置
lb = -5.12*ones(1,10);  % 10维搜索空间
ub = 5.12*ones(1,10);
NP = 50; max_iter = 1000;

% 运行算法
[best_sol, best_fit, history] = AdaptiveDE(rastrigin, lb, ub, 10, NP, max_iter);

% 可视化结果
figure;
plot(history);
xlabel('迭代次数'); ylabel('最优值');
title('Rastrigin函数优化过程');

2. Ackley函数测试

ackley = @(x) -20*exp(-0.2*sqrt(mean(x.^2))) - exp(mean(cos(2*pi*x))) + 20 + exp(1);

lb = -32.768*ones(1,5);
ub = 32.768*ones(1,5);
[best_sol, best_fit] = AdaptiveDE(ackley, lb, ub, 5, 100, 500);
disp(['最优解: ', num2str(best_sol)]);
disp(['最优值: ', num2str(best_fit)]);

参考代码 自适应差分进化算法例程 www.youwenfan.com/contentcnh/59497.html

五、性能优化技巧

1. 边界处理：采用反射边界策略

   U = max(U, lb);
   U = min(U, ub);
   

2. 精英保留：始终保留当前最优解

   if new_fit < best_fit
       best_sol = U;
       best_fit = new_fit;
   end
   

3. 并行计算：使用parfor加速种群评估

   parfor i = 1:NP
       fitness(i) = func(pop(i,:));
   end
   

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六、应用场景示例

1. 工程优化问题

% 压力容器设计优化
pressure_vessel = @(x) 0.6224*x(1)*x(2)*x(3) + 1.7781*x(1)^2*x(3) + ...
                     3.1661*x(2)^2*x(3) + 19.84*x(1)*x(2);

lb = [0,0,0,0]; ub = [100,100,100,100];
[best_sol, best_fit] = AdaptiveDE(pressure_vessel, lb, ub, 4, 60, 1000);

2. 无人机路径规划

% 三维路径规划目标函数
path_cost = @(x) sum(x.^2) + 10*max(0, x(3)-50);

lb = [-100,-100,-100]; ub = [100,100,100];
[best_sol, best_fit] = AdaptiveDE(path_cost, lb, ub, 3, 30, 500);