ADMM筆記_basis Pursuit 代碼重點詳解

源代碼出自boyd

https://web.stanford.edu/~boyd/papers/admm/basis_pursuit/basis_pursuit.html

給出詳細備註

function [z, history] = basis_pursuit(A, b, rho, alpha)
     % 解決如下 ADMM問題:
     %   minimize     ||x||_1
     %   subject to   Ax = b
     %其中返回的 history變量包含 目標值、原始殘差和對偶殘差，以及每次迭代時原始殘差和對偶殘差的容差（the objective %value, the primal and  dual residual norms, and the tolerances for the primal and dual residual  norms at each iteration.）
     % rho is the augmented Lagrangian parameter.   
     % alpha is the over-relaxation parameter (typical values for alpha are between 1.0 and 1.8).

     t_start = tic;
     % Global constants and defaults
     QUIET    = 0;
     MAX_ITER = 1000;
     ABSTOL   = 1e-4;
     RELTOL   = 1e-2;     
     %% Data preprocessing
     [m n] = size(A);
     %% ADMM solver

基本上是按paper中寫的順序來的

     x = zeros(n,1);
     z = zeros(n,1);
     u = zeros(n,1); 
     if ~QUIET
         fprintf('%3s\t%10s\t%10s\t%10s\t%10s\t%10s\n', 'iter', ...
           'r norm', 'eps pri', 's norm', 'eps dual', 'objective');
     end       %不斷輸出狀態量
     % precompute static variables for x-update (projection on to Ax=b)
     AAt = A*A';
     P = eye(n) - A' * (AAt \ A);
     q = A' * (AAt \ b);
     
     for k = 1:MAX_ITER   %迭代
         % x-update
         x = P*(z - u) + q;
     
         % z-update with relaxation
         zold = z;
         x_hat = alpha*x + (1 - alpha)*zold; 

%x這兒做了一個over-relaxation

         z = shrinkage(x_hat + u, 1/rho);

%shrinkage是一個soft thresholding  operator,如下所示：

     
         u = u + (x_hat - z);
     
         % diagnostics, reporting, termination checks
         history.objval(k)  =norm(x,1);    %目標函數
     
         history.r_norm(k)  = norm(x - z);     
         history.s_norm(k)  = norm(-rho*(z - zold));
         
         history.eps_pri(k) = sqrt(n)*ABSTOL + RELTOL*max(norm(x), norm(-z));
         history.eps_dual(k)= sqrt(n)*ABSTOL + RELTOL*norm(rho*u);

%這邊是如下的一種終止條件，
     
         if ~QUIET
             fprintf('%3d\t%10.4f\t%10.4f\t%10.4f\t%10.4f\t%10.2f\n', k, ...
                 history.r_norm(k), history.eps_pri(k), ...
                 history.s_norm(k), history.eps_dual(k), history.objval(k));
         end
     
         if (history.r_norm(k) < history.eps_pri(k) && ...
            history.s_norm(k) < history.eps_dual(k))
              break;
         end
     end
     if ~QUIET
         toc(t_start);
     end
     end
     
     function y = shrinkage(a, kappa)
         y = max(0, a-kappa) - max(0, -a-kappa);
     end