粒子群优化算法 (PSO) 寻找 Shubert 函数全局最优解

本代码使用粒子群优化算法 (PSO) 寻找 Shubert 函数的全局最优解，并输出最优解的坐标和函数值。

1. PSO 算法实现

function [xopt, yopt, fopt] = PSO(objfun, np, maxiter, bounds)
    % 初始化粒子
    x = rand(np, 1) * (bounds(2, 1) - bounds(1, 1)) + bounds(1, 1);
    y = rand(np, 1) * (bounds(2, 2) - bounds(1, 2)) + bounds(1, 2);
    v = zeros(np, 2);
    pbestx = x;
    pbesty = y;
    pbestf = inf(np, 1);
    for i = 1:np
        f = objfun(x(i), y(i));
        if f < pbestf(i)
            pbestf(i) = f;
        end
    end
    gbesti = find(pbestf == min(pbestf));
    gbestx = pbestx(gbesti);
    gbesty = pbesty(gbesti);
    gbestf = pbestf(gbesti);

    % 迭代优化
    for iter = 1:maxiter
        w = 0.5; % 惯性权重
        c1 = 2; % 个体学习因子
        c2 = 2; % 社会学习因子
        for i = 1:np
            r1 = rand;
            r2 = rand;
            v(i, 1) = w * v(i, 1) + c1 * r1 * (pbestx(i) - x(i)) + c2 * r2 * (gbestx - x(i));
            x(i) = x(i) + v(i, 1);
            if x(i) < bounds(1, 1)
                x(i) = bounds(1, 1);
                v(i, 1) = -v(i, 1);
            elseif x(i) > bounds(2, 1)
                x(i) = bounds(2, 1);
                v(i, 1) = -v(i, 1);
            end

            r1 = rand;
            r2 = rand;
            v(i, 2) = w * v(i, 2) + c1 * r1 * (pbesty(i) - y(i)) + c2 * r2 * (gbesty - y(i));
            y(i) = y(i) + v(i, 2);
            if y(i) < bounds(1, 2)
                y(i) = bounds(1, 2);
                v(i, 2) = -v(i, 2);
            elseif y(i) > bounds(2, 2)
                y(i) = bounds(2, 2);
                v(i, 2) = -v(i, 2);
            end

            f = objfun(x(i), y(i));
            if f < pbestf(i)
                pbestf(i) = f;
                pbestx(i) = x(i);
                pbesty(i) = y(i);
            end
        end

        [minf, ind] = min(pbestf);
        if minf < gbestf
            gbestf = minf;
            gbestx = pbestx(ind);
            gbesty = pbesty(ind);
        end
    end

    % 返回最优解及其函数值
    xopt = gbestx;
    yopt = gbesty;
    fopt = gbestf;
end

2. Shubert 函数定义

function f = Shubert(x, y)
    A = 0;
    B = 0;
    for i = 1:5
        A = A + i * cos((i + 1) * x + i);
        B = B + i * cos((i + 1) * y + i);
    end
    f = A * B;
end

3. 运行程序

% 设置搜索范围和粒子数量
bounds = [-10, -10; 10, 10];
np = 50;
maxiter = 1000;

% 运行PSO算法寻找最优解
[xopt, yopt, fopt] = PSO(@Shubert, np, maxiter, bounds);

% 输出结果
fprintf('最优解为：x = %f， y = %f， f = %f\n', xopt, yopt, fopt);

注意： Shubert 函数有多个局部最优解，PSO 算法无法保证找到全局最优解，需要多次运行并比较结果。为了找到更多局部最优解，可以尝试以下方法:

增加粒子数量和迭代次数
设置多个起始点并分别运行 PSO 算法
使用其他全局优化算法，例如遗传算法 (GA) 或模拟退火算法 (SA)

通过上述方法，可以更好地探索 Shubert 函数的搜索空间，并找到更多局部最优解。