admm算法matlab代码

以下是ADMM算法的Matlab代码示例：

function [x, history] = admm(A, b, rho, alpha)
% ADMM solver for the following problem:
%
%   minimize || Ax - b ||_2^2 + alpha || x ||_1
%
% The input parameters are:
% A     - a matrix
% b     - a vector
% rho   - the penalty parameter
% alpha - the regularization parameter
%
% The output values are:
% x       - the solution of the problem
% history - a struct containing the objective value and other information

[m, n] = size(A);

% ADMM solver
x = zeros(n, 1);
z = zeros(n, 1);
u = zeros(n, 1);

% Precompute
L = chol(2 * rho * (A' * A) + eye(n), 'lower');

% Start ADMM iterations
MAX_ITER = 1000;
ABSTOL = 1e-4;
RELTOL = 1e-2;

for iter = 1:MAX_ITER

    % x-update
    q = A' * b + rho * (z - u);    % temporary value
    x = L \ (L' \ q);

    % z-update with relaxation
    zold = z;
    x_hat = alpha / rho * x + (1 - alpha / rho) * zold;
    z = shrinkage(x_hat + u, 1 / rho);

    % u-update
    u = u + (x_hat - z);

    % diagnostics, reporting, termination checks
    history.objval(iter) = objective(A, b, alpha, x, z);

    history.r_norm(iter) = norm(x - z);
    history.s_norm(iter) = norm(-rho * (z - zold));

    history.eps_pri(iter) = sqrt(n) * ABSTOL + RELTOL * max(norm(x), norm(-z));
    history.eps_dual(iter) = sqrt(n) * ABSTOL + RELTOL * norm(rho * u);

    if (history.r_norm(iter) < history.eps_pri(iter) && ...
       history.s_norm(iter) < history.eps_dual(iter))
         break;
    end

end

end

function obj = objective(A, b, alpha, x, z)
obj = norm(A * x - b)^2 / 2 + alpha * norm(z, 1);
end

function y = shrinkage(x, kappa)
y = max(0, x - kappa) - max(0, -x - kappa);
end

此代码实现了ADMM求解以下问题：

$$\min_{x}\frac{1}{2}\|Ax-b\|_2^2+\alpha\|x\|_1$$

其中，$A$是一个$m\times n$的矩阵，$b$是一个$m$维向量，$\alpha$是正则化参数，$x$是一个$n$维向量。