蚁群优化算法matlab代码，带测试函数

以下是蚁群优化算法的 Matlab 代码，同时附带一个测试函数 Sphere Function：

% Sphere Function
function f = sphere(x)
% x: n-dimensional vector
f = sum(x.^2);

% Ant Colony Optimization Algorithm
function [x, fval] = ant_colony_optimization(n, L, m, q0, alpha, beta, rho, tau0, tmax)
% n: dimension of the search space
% L: search space limit
% m: number of ants
% q0: probability of selecting the best known solution
% alpha: importance of pheromone trail
% beta: importance of heuristic information
% rho: pheromone trail persistence factor
% tau0: initial pheromone trail
% tmax: maximum number of iterations

% Initialize pheromone trail
tau = tau0 * ones(n+1, L+1);

% Initialize best solution
x_best = rand(1, n) * 2 * L - L;
f_best = sphere(x_best);

for t = 1:tmax
    % Initialize ant positions
    x = rand(m, n) * 2 * L - L;
    f = zeros(m, 1);
    
    % Ant movement
    for i = 1:m
        for j = 1:n
            % Calculate transition probability
            p = tau(j, round(x(i,j)+L)+1).^alpha .* (1./abs(x(i,j)-x_best(j))).^beta;
            p = p / sum(p);
            
            % Choose next position
            if rand < q0
                [~, idx] = max(p);
            else
                idx = roulette_wheel(p);
            end
            x(i,j) = x(i,j) + idx - L - 1;
        end
        
        % Evaluate solution
        f(i) = sphere(x(i,:));
        
        % Update best known solution
        if f(i) < f_best
            x_best = x(i,:);
            f_best = f(i);
        end
    end
    
    % Update pheromone trail
    delta_tau = zeros(n+1, L+1);
    for i = 1:m
        for j = 1:n
            delta_tau(j, round(x(i,j)+L)+1) = delta_tau(j, round(x(i,j)+L)+1) + 1/f(i);
        end
        delta_tau(n+1, round(f(i))+1) = delta_tau(n+1, round(f(i))+1) + 1/f(i);
    end
    tau = (1-rho) * tau + delta_tau;
end

% Return best solution
x = x_best;
fval = f_best;

% Roulette wheel selection
function idx = roulette_wheel(p)
r = rand;
c = cumsum(p);
idx = find(r <= c, 1, 'first');

% Example usage
n = 10;
L = 10;
m = 50;
q0 = 0.9;
alpha = 1;
beta = 2;
rho = 0.5;
tau0 = 0.1;
tmax = 100;
[x, fval] = ant_colony_optimization(n, L, m, q0, alpha, beta, rho, tau0, tmax);