function [x,fval,stat] = gnnls(H,f,options)% GNNLS Solves Generalized Non-negative Least Squares Problem (GNNLS). %% Synopsis:%  [x,fval,stat] = gnnls(H,f)%  [x,fval,stat] = gnnls(H,f,options)%% Description:%  The Generalized Non-negative Least Squares to solve reads %%   argmin 0.5*x'*H*x + f'*x  subject to: x(i) >= 0 for all i.%     x%%  H is symetric positive semi-definite matrix and f a vector. %  The GNNLS is a special instance of the Quadratig Programming %  (QP) task which can be solved by the Sequential Coordinate-wise %  Algorithm (Franc, Hlavac: Sequential Coordinate-wise Algorithm%  for Non-negative Least Squares Problem. Research report. %  CTU-CMP-2005-06. CTU FEL Prague. %  ftp://cmp.felk.cvut.cz/pub/cmp/articles/franc/Franc-TR-2005-06.ps).%%  The optimization halts if one of the following stopping %  conditions is satisfied:%%    t >= options.tmax           ->  exit_flag = 0  Number of iterations.%    UB-LB <= options.tolabs     ->  exit_flag = 1  Abs. tolerance.%    UB-LB <= UB*options.tolrel  ->  exit_flag = 2  Relative tolerance.%    Relaxed KKT cond. satisfied ->  exit_flag = 3  It means that%       mu(i) >= -options.tolKKT  for all i &&  %       mu(i) <= options.tolKKT   for i such that x(i) > 0%       where mu = H*x + f                        %%  where t is number of iterations, UB and LB are upper and %  lower bounds on the optimal solution. The stopping conditions %  using UB and LB can be applied only if the matric H has all elements %  non-negative.%% Input:%  H [dim x dim] Symetric positive definite matrix.%  f [dim x 1] Vector.%  options [struct] Control parameters:%   .tmax [1x1] Maximal number of iterations (default inf).%   .tolabs [1x1] Absolute tolerance stopping condition (default 0.0).%   .tolrel [1x1] Relative tolerance stopping condition (default 1e-6).%   .tolKKT [1x1] Tolerance of satisfaction of KKT conditions (default 1e-6).%   .verb [1x1] If > 0 then displays info every verb-th iteration (default 0).%% Output:%  x [dim x 1] Solution vector.%  fval [1x1] Value of the criterion at x.%  stat [struct] Constains:%   .exiflag [1x1] Exitflag (see above).%   .t [1x1] Number of iterations.%   .LB_History [1 x t] History of lower bounds.%   .UB_History [1 x t] History of upper bounds.%   .access [1x1] Number of accesses to the matrox H.%   .LB [1x1] == LB_History(end).%   .UB [1x1] == UB_History(end).%   .cputime [1x1] CPU time meassured by tic toc functions.%% Example:%  A = rand(1000,250); b = rand(1000,1); %  tic; x1 = lsqnonneg(A,b); toc, F1 = norm(A*x1-b)^2%  tic; x2 = gnnls(A'*A, -A'*b); toc, F2 = norm(A*x2-b)^2%% See also %  GMNP.%% About: Statistical Pattern Recognition Toolbox% (C) 1999-2005, Written by Vojtech Franc and Vaclav Hlavac% <a href="http://www.cvut.cz">Czech Technical University Prague</a>% <a href="http://www.feld.cvut.cz">Faculty of Electrical Engineering</a>% <a href="http://cmp.felk.cvut.cz">Center for Machine Perception</a>% Modifications:% 09-sep-2005, VF% 28-jan-2005, VF% cputimertic;% optionsif nargin < 3, options = []; else options = c2s(options); endif ~isfield(options,'tolabs'), options.tolabs = 0; endif ~isfield(options,'tolrel'), options.tolrel = 0; endif ~isfield(options,'tolKKT'), options.tolKKT = 1e-6; endif ~isfield(options,'tmax'), options.tmax = inf; endif ~isfield(options,'verb'), options.verb = 0; endif ~isfield(options,'solver'), options.solver = 'sca'; end[x,exitflag,t,access,History] = gnnls_mex(H,f,options.solver,...   options.tmax,options.tolabs,options.tolrel,options.tolKKT,options.verb);stat.t = t;stat.exitflag = exitflag;stat.LB_History = History(1,:);stat.UB_History = History(2,:);stat.LB = History(1,end);stat.UB = History(2,end);stat.access = access;stat.cputime = toc;fval = stat.UB;return;% EOF