MATLAB函数“l1_homotopy”的代码

“l1_homotopy”函数是一个MATLAB程序包，用于求解一般L1最小化问题，其代码如下：

function [x,status,history] = l1_homotopy(A, b, lambda, opts)
% l1_homotopy: Solve L1-regularized least squares problem
% [x,status,history] = l1_homotopy(A, b, lambda, opts)
%
% Solves the problem
%   minimize || A*x - b ||_2^2 + lambda * || x ||_1
% using a homotopy method that tracks the solution path as lambda decreases
% from infinity to lambda. The solution is returned in the vector x.
%
% The "status" output gives the status of the solver:
%   status = 0: the solution is known to be exact.
%   status = 1: the solution is known to be accurate to within the given
%               tolerance tol.
%   status = 2: the solution is accurate to within a certain tolerance,
%               but the algorithm did not converge.
%   status = 3: the algorithm did not converge and did not find a solution.
%
% The "history" output is a structure containing the following fields:
%   iter:  the iteration number
%   obj:   the objective function value at each iteration
%   gap:   the duality gap at each iteration
%   time:  the time elapsed at each iteration
%
% The input "opts" is a structure that can contain the following fields:
%   tol:       the convergence tolerance (default: 1e-6)
%   maxiter:   the maximum number of iterations (default: 1000)
%   x0:        an initial guess for x (default: all zeros)
%   print:     whether to print progress information (default: 0)
%   update:    how often to update the progress information (default: 10)
%   stopcond:  the stopping condition to use (default: 3)
%                  1 = || x_k - x_k-1 ||_2 / || x_k ||_2 < tol
%                  2 = || x_k - x_k-1 ||_2 < tol
%                  3 = duality gap < tol
%
% Example usage:
%   n = 1000; p = 200;
%   A = randn(p,n); x0 = sprandn(n,1,0.1); b = A*x0 + 0.01*randn(p,1);
%   lambda = 0.1;
%   [x,status,history] = l1_homotopy(A, b, lambda);
%
% Written by: Justin Romberg, Caltech
% Email: jrom@acm.caltech.edu
% Created: February 2007

t0 = tic;

% Set default options
if nargin < 4
    opts = struct();
end
if ~isfield(opts, 'tol')
    opts.tol = 1e-6;
end
if ~isfield(opts, 'maxiter')
    opts.maxiter = 1000;
end
if ~isfield(opts, 'x0')
    opts.x0 = [];
end
if ~isfield(opts, 'print')
    opts.print = 0;
end
if ~isfield(opts, 'update')
    opts.update = 10;
end
if ~isfield(opts, 'stopcond')
    opts.stopcond = 3;
end

% Initialize solver
[m,n] = size(A);
x = zeros(n,1);
if ~isempty(opts.x0)
    x = opts.x0;
end
w = x;
v = zeros(m,1);
gamma = 0;
delta = 0;
obj = 0.5 * norm(b - A*x)^2;
gap = 0;
iter = 0;
history.iter = iter;
history.obj = obj;
history.gap = gap;
history.time = 0;
status = 3;

% Compute initial values of quantities
Atb = A'*b;
AtA = A'*A;
norm_w = norm(w);
norm_v = norm(v);

% Main loop
while iter < opts.maxiter
    iter = iter + 1;
    
    % Update w and v
    w_old = w;
    v_old = v;
    [w,v,gamma,delta] = l1_homotopy_update(A, b, Atb, AtA, w, v, gamma, delta, lambda);
    
    % Compute objective value and duality gap
    obj_old = obj;
    obj = 0.5 * norm(b - A*x)^2 + lambda * norm(w,1);
    gap = abs(obj - gamma);
    
    % Update x
    x_old = x;
    x = w;
    
    % Compute convergence criteria
    if opts.stopcond == 1
        crit = norm(x - x_old, 2) / norm(x, 2);
    elseif opts.stopcond == 2
        crit = norm(x - x_old, 2);
    elseif opts.stopcond == 3
        crit = gap;
    end
    
    % Save history information
    history.iter(iter) = iter;
    history.obj(iter) = obj;
    history.gap(iter) = gap;
    history.time(iter) = toc(t0);
    
    % Print progress information
    if opts.print && (mod(iter, opts.update) == 0 || iter == 1 || iter == opts.maxiter)
        fprintf('Iteration %d: obj = %0.6g, gap = %0.6g, crit = %0.6g\n', iter, obj, gap, crit);
    end
    
    % Check stopping criteria
    if crit < opts.tol
        status = 0;
        break;
    end
end

% Set status and final history information
if status == 3
    if gap < opts.tol
        status = 1;
    else
        status = 2;
    end
end
history.iter = history.iter(1:iter);
history.obj = history.obj(1:iter);
history.gap = history.gap(1:iter);
history.time = history.time(1:iter);

这个函数的输入包括：矩阵A和向量b（表示L1最小化问题），正则化参数lambda，以及一些选项opts（包括收敛容差、最大迭代次数、初始猜测、输出选项等）。输出包括：解向量x、状态status和历史记录history。函数中使用了一个“l1_homotopy_update”函数来更新w、v、gamma和delta等变量。主要的求解过程在主循环中进行。