利用MATLAB编写代码实现CMA算法

CMA算法是一种常见的优化算法，用于解决连续优化问题。下面是MATLAB代码实现CMA算法的示例：

function [x, fval] = cma(fitnessfcn, x0, sigma0, options)
% fitnessfcn: 目标函数句柄
% x0: 初始值
% sigma0: 初始步长
% options: 选项

% 初始化参数
N = length(x0);
lambda = 4 + floor(3*log(N));
mu = floor(lambda/2);
weights = log(mu+1/2)-log(1:mu)';
weights = weights/sum(weights);
mueff = sum(weights)^2/sum(weights.^2);

cc = (4+mueff/N)/(N+4+2*mueff/N);
cs = (mueff+2)/(N+mueff+5);
c1 = 2/((N+1.3)^2+mueff);
cmu = min(1-c1, 2*(mueff-2+1/mueff)/((N+2)^2+mueff));
damps = 1+2*max(0, sqrt((mueff-1)/(N+1))-1)+cs;

pc = zeros(N,1);
ps = zeros(N,1);
B = eye(N);
D = ones(N,1);
C = B*diag(D.^2)*B';

% 开始迭代
x = x0;
sigma = sigma0;
evals = 0;
[fval, ~] = feval(fitnessfcn, x);
bestfval = fval;
bestx = x;

for iter = 1:options.MaxIter
    % 生成子代
    pop = repmat(x,1,lambda) + sigma*(B*diag(D)*randn(N,lambda));
    % 计算适应度
    [fvals, idx] = sort(feval(fitnessfcn, pop));
    evals = evals + lambda;
    % 更新最优解
    if fvals(1) < bestfval
        bestfval = fvals(1);
        bestx = pop(:,idx(1));
    end
    % 更新权重
    xold = x;
    x = pop(:,idx(1:mu))*weights;
    % 更新进化路径
    ps = (1-cs)*ps + sqrt(cs*(2-cs)*mueff)*(B*x);
    hsig = norm(ps)/sqrt(1-(1-cs)^(2*evals/lambda))/chiN(N) < 1.4+2/(N+1);
    pc = (1-cc)*pc + hsig*sqrt(cc*(2-cc)*mueff)*(B*x);
    % 更新协方差矩阵
    artmp = (1/sigma)*(pop(:,idx(1:mu))-repmat(xold,1,mu));
    C = (1-c1-cmu)*C + c1*(pc*pc'+(1-hsig)*cc*(2-cc)*C) + cmu*artmp*diag(weights)*artmp';
    % 更新步长
    sigma = sigma*exp((cs/damps)*(norm(ps)/chiN(N)-1));
    % 更新B和D
    if evals/N/lambda > 1/10
        D = diag(sqrt(diag(C)));
        B = orth(C);
    end
    % 检查终止条件
    if sigma*max(D) < options.TolFun && max(abs(x-xold)) < options.TolX
        break;
    end
end

x = bestx;
fval = bestfval;
end

function y = chiN(N)
y = sqrt(N)*(1-1/(4*N)+1/(21*N^2));
end

其中，`fitnessfcn`是自己定义的目标函数句柄，`x0`是初始值，`sigma0`是初始步长，`options`是选项结构体，包含最大迭代次数`MaxIter`、目标函数值和解的精度`TolFun`和`TolX`等。函数返回最优解`x`和最优解对应的目标函数值`fval`。