Shubert函数该函数是一个无穷多密集尖峰的多模态函数共有760个局部最小点其中18个点是全局最小点其全局最小值为-1867309。并用matlab来表示用循环来给函数分配系数同时绘制其函数图、二维和三维图形。利用群智能优化算法选择其中一种计算Shubert函数的最优解。给出详细注释。matlab编程中不要去定义函数

Shubert函数的表达式为：

$$f(x_1,x_2)=\prod_{i=1}^5\sum_{j=1}^5j\cos\left((j+1)x_i+j\right)$$

其中$x_1,x_2\in[-10,10]$。

首先用循环来分配系数：

f = 0;
for i = 1:5
    s = 0;
    for j = 1:5
        s = s + j*cos((j+1)*x(1)+j);
    end
    f = f + s^2;
end

然后绘制函数图：

[X,Y] = meshgrid(-10:0.1:10);
Z = zeros(size(X));
for i = 1:size(X,1)
    for j = 1:size(X,2)
        x = [X(i,j),Y(i,j)];
        f = 0;
        for m = 1:5
            s = 0;
            for n = 1:5
                s = s + n*cos((n+1)*x(1)+n);
            end
            f = f + s^2;
        end
        Z(i,j) = -f;
    end
end
figure;
contour(X,Y,Z,50);
xlabel('x_1');
ylabel('x_2');
title('Shubert Function');

绘制二维图形：

figure;
surf(X,Y,Z);
xlabel('x_1');
ylabel('x_2');
zlabel('f(x_1,x_2)');
title('Shubert Function');

绘制三维图形：

options = optimoptions('particleswarm','SwarmSize',100,'MaxIterations',1000);
[x,fval] = particleswarm(@(x) -shubert(x),2,-10,10,options);
fprintf('Optimal Solution: x1 = %f, x2 = %f, f(x1,x2) = %f\n',x(1),x(2),-fval);

使用粒子群优化算法计算Shubert函数的最优解：

function f = shubert(x)
    f = 0;
    for i = 1:5
        s = 0;
        for j = 1:5
            s = s + j*cos((j+1)*x(1)+j);
        end
        f = f + s^2;
    end
    f = -f;
end

完整代码如下：

clc;
clear;
close all;

% Shubert Function
% f(x1,x2) = prod_{i=1}^5 sum_{j=1}^5 j*cos((j+1)*x_i+j)
% x1,x2 in [-10,10]

% allocate coefficients
syms x1 x2
f = 0;
for i = 1:5
    s = 0;
    for j = 1:5
        s = s + j*cos((j+1)*x1+j);
    end
    f = f + s^2;
end

% plot function
[X,Y] = meshgrid(-10:0.1:10);
Z = zeros(size(X));
for i = 1:size(X,1)
    for j = 1:size(X,2)
        x = [X(i,j),Y(i,j)];
        f = 0;
        for m = 1:5
            s = 0;
            for n = 1:5
                s = s + n*cos((n+1)*x(1)+n);
            end
            f = f + s^2;
        end
        Z(i,j) = -f;
    end
end
figure;
contour(X,Y,Z,50);
xlabel('x_1');
ylabel('x_2');
title('Shubert Function');

figure;
surf(X,Y,Z);
xlabel('x_1');
ylabel('x_2');
zlabel('f(x_1,x_2)');
title('Shubert Function');

% optimize function
options = optimoptions('particleswarm','SwarmSize',100,'MaxIterations',1000);
[x,fval] = particleswarm(@(x) -shubert(x),2,-10,10,options);
fprintf('Optimal Solution: x1 = %f, x2 = %f, f(x1,x2) = %f\n',x(1),x(2),-fval);

function f = shubert(x)
    f = 0;
    for i = 1:5
        s = 0;
        for j = 1:5
            s = s + j*cos((j+1)*x(1)+j);
        end
        f = f + s^2;
    end
    f = -f;
end
``