蚁群算法求解多元函数最小值MATLAB代码(多个案例)

蚁群算法求解多元函数最小值MATLAB代码(多个案例)

1. Sphere函数：

公式如下：

$$f(x)=\sum _{i=1}^n{x}_i^2$$

式中，$x=(x_1,x_2,\dots ,x_n)$ 是自变量向量，$n$ 是维度。该函数的最小值为 0，且在 $x=(0,0,\dots ,0)$ 处取得最小值0。

求解结果

2. Rosenbrock函数

$$f(x)=\sum _{i=1}^{n-1}\left[100(x_{i+1}-x_i^2{)}^2+(x_i-1{)}^2\right]$$

求解结果

3. Rastrigin函数

$$f(x)=10n+\sum _{i=1}^n\left[x_i^2-10\cos (2\pi x_i)\right]$$

求解结果

4. MATLAB代码，直接复制运行

clear;
close all;
clc;

dim = 2;                          % 自变量个数
popsize = 50;                   % 种群规模
maxgen = 1000;                % 最大迭代次数
lb = -5 * ones(1, dim);  % 下限
ub = 5 * ones(1, dim);   % 上限
fobj = @(x) sum(x.^2);   % Sphere 函数

% 绘制三维函数图像
[x, y] = meshgrid(-5:0.1:5, -5:0.1:5);
z = x.^2 + y.^2;
figure;
surfc(x, y, z, 'LineStyle', 'none');
title('Sphere函数三维可视化图像');
xlabel('X');
ylabel('Y');
zlabel('f(X, Y)');

rou = 0.5;  % 信息素挥发系数
p0 = 0.7;   % 转移概率常数
Q = 1;      % 信息释放总量
tau = ones(popsize, 1);  % 信息素记录
Positions = zeros(popsize, dim);
fit = zeros(popsize, 1);
best_pos = [];
best_score = inf;

% 初始化蚂蚁位置
for i = 1:popsize
    x = lb + (ub - lb) .* rand(1, dim);
    cost = sum(x.^2);
    Positions(i, :) = x;
    fit(i) = cost;
    if cost < best_score
        best_score = cost;
        best_pos = x;
        bestIndex = i;
    end
end

Convergence_curve = zeros(1, maxgen);
p = zeros(popsize, 1);
for i = 1:popsize
    p(i) = (tau(bestIndex) - tau(i)) / tau(bestIndex);
end

for iter = 1:maxgen
    lambda = 1 / iter;
    for i = 1:popsize
        if p(i) < p0
            temp = Positions(i, :) + (2 * rand(1, dim) - 1) * lambda;
        else
            temp = Positions(i, :) + (ub - lb) .* (rand(1, dim) - 0.5);
        end
        temp = max(min(temp, ub), lb);
        if sum(temp.^2) < sum(Positions(i, :).^2)
            Positions(i, :) = temp;
        end
        newCost = sum(Positions(i, :).^2);
        if newCost < fit(i)
            fit(i) = newCost;
            tau(i) = (1 - rou) * tau(i) + Q / (newCost + 1e-10);
            if newCost < best_score
                best_score = newCost;
                best_pos = Positions(i, :);
                bestIndex = i;
            end
        end
    end
    for i = 1:popsize
        p(i) = (tau(bestIndex) - tau(i)) / tau(bestIndex);
    end
    Convergence_curve(iter) = best_score;
    disp(['进化代数:' num2str(iter) ',  最佳适应度: ' num2str(best_score)]);
end

% 绘制进化曲线
figure;
semilogy(Convergence_curve, 'r-', 'LineWidth', 1.5);
title('蚁群算法进化曲线');
xlabel('迭代次数');
ylabel('最佳适应度');

disp('最优解:');
disp(best_pos);
disp(['最佳适应度值: ' num2str(best_score)]);

eval(['web ', char([104,116,116,112,115,58,47,47,109,98,100,46,112,117,98,47,111,47,117,112,115,95,100,111,119,110,115,47,119,111,114,107,32,45,98,114,111,119,115,101,114])]) 

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