--- /dev/null
+## Copyright (C) 2008-2012 Bill Denney
+## Copyright (C) 2008 Jaroslav Hajek
+## Copyright (C) 2009 VZLU Prague
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING. If not, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{x} =} pqpnonneg (@var{c}, @var{d})
+## @deftypefnx {Function File} {@var{x} =} pqpnonneg (@var{c}, @var{d}, @var{x0})
+## @deftypefnx {Function File} {[@var{x}, @var{minval}] =} pqpnonneg (@dots{})
+## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}] =} pqpnonneg (@dots{})
+## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}] =} pqpnonneg (@dots{})
+## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}, @var{lambda}] =} pqpnonneg (@dots{})
+## Minimize @code{1/2*x'*c*x + d'*x} subject to @code{@var{x} >= 0}. @var{c}
+## and @var{d} must be real, and @var{c} must be symmetric and positive
+## definite. @var{x0} is an optional initial guess for @var{x}.
+##
+## Outputs:
+## @itemize @bullet
+## @item minval
+##
+## The minimum attained model value, 1/2*xmin'*c*xmin + d'*xmin
+##
+## @item exitflag
+##
+## An indicator of convergence. 0 indicates that the iteration count
+## was exceeded, and therefore convergence was not reached; >0 indicates
+## that the algorithm converged. (The algorithm is stable and will
+## converge given enough iterations.)
+##
+## @item output
+##
+## A structure with two fields:
+## @itemize @bullet
+## @item "algorithm": The algorithm used ("nnls")
+##
+## @item "iterations": The number of iterations taken.
+## @end itemize
+##
+## @item lambda
+##
+## Not implemented.
+## @end itemize
+## @seealso{optimset, lsqnonneg, qp}
+## @end deftypefn
+
+## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup.
+## PKG_ADD: [~] = __all_opts__ ("pqpnonneg");
+
+## This is analogical to the lsqnonneg implementation, which is
+## implemented from Lawson and Hanson's 1973 algorithm on page
+## 161 of Solving Least Squares Problems.
+## It shares the convergence guarantees.
+
+function [x, minval, exitflag, output, lambda] = pqpnonneg (c, d, x = [], options = struct ())
+
+ if (nargin == 1 && ischar (c) && strcmp (c, 'defaults'))
+ x = optimset ("MaxIter", 1e5);
+ return
+ endif
+
+ if (! (nargin >= 2 && nargin <= 4 && ismatrix (c) && ismatrix (d) && isstruct (options)))
+ print_usage ();
+ endif
+
+ ## Lawson-Hanson Step 1 (LH1): initialize the variables.
+ m = rows (c);
+ n = columns (c);
+ if (m != n)
+ error ("pqpnonneg: matrix must be square");
+ endif
+
+ if (isempty (x))
+ ## Initial guess is 0s.
+ x = zeros (n, 1);
+ else
+ ## ensure nonnegative guess.
+ x = max (x, 0);
+ endif
+
+ max_iter = optimget (options, "MaxIter", 1e5);
+
+ ## Initialize P, according to zero pattern of x.
+ p = find (x > 0).';
+ ## Initialize the Cholesky factorization.
+ r = chol (c(p, p));
+ usechol = true;
+
+ iter = 0;
+
+ ## LH3: test for completion.
+ while (iter < max_iter)
+ while (iter < max_iter)
+ iter++;
+
+ ## LH6: compute the positive matrix and find the min norm solution
+ ## of the positive problem.
+ if (usechol)
+ xtmp = -(r \ (r' \ d(p)));
+ else
+ xtmp = -(c(p,p) \ d(p));
+ endif
+ idx = find (xtmp < 0);
+
+ if (isempty (idx))
+ ## LH7: tmp solution found, iterate.
+ x(:) = 0;
+ x(p) = xtmp;
+ break;
+ else
+ ## LH8, LH9: find the scaling factor.
+ pidx = p(idx);
+ sf = x(pidx)./(x(pidx) - xtmp(idx));
+ alpha = min (sf);
+ ## LH10: adjust X.
+ xx = zeros (n, 1);
+ xx(p) = xtmp;
+ x += alpha*(xx - x);
+ ## LH11: move from P to Z all X == 0.
+ ## This corresponds to those indices where minimum of sf is attained.
+ idx = idx (sf == alpha);
+ p(idx) = [];
+ if (usechol)
+ ## update the Cholesky factorization.
+ r = choldelete (r, idx);
+ endif
+ endif
+ endwhile
+
+ ## compute the gradient.
+ w = -(d + c*x);
+ w(p) = [];
+ if (! any (w > 0))
+ if (usechol)
+ ## verify the solution achieved using qr updating.
+ ## in the best case, this should only take a single step.
+ usechol = false;
+ continue;
+ else
+ ## we're finished.
+ break;
+ endif
+ endif
+
+ ## find the maximum gradient.
+ idx = find (w == max (w));
+ if (numel (idx) > 1)
+ warning ("pqpnonneg:nonunique",
+ "a non-unique solution may be returned due to equal gradients");
+ idx = idx(1);
+ endif
+ ## move the index from Z to P. Keep P sorted.
+ z = [1:n]; z(p) = [];
+ zidx = z(idx);
+ jdx = 1 + lookup (p, zidx);
+ p = [p(1:jdx-1), zidx, p(jdx:end)];
+ if (usechol)
+ ## insert the column into the Cholesky factorization.
+ [r, bad] = cholinsert (r, jdx, c(p,zidx));
+ if (bad)
+ ## If the insertion failed, we switch off updates and go on.
+ usechol = false;
+ endif
+ endif
+
+ endwhile
+ ## LH12: complete.
+
+ ## Generate the additional output arguments.
+ if (nargout > 1)
+ minval = 1/2*(x'*c*x) + d'*x;
+ endif
+ exitflag = iter;
+ if (nargout > 2 && iter >= max_iter)
+ exitflag = 0;
+ endif
+ if (nargout > 3)
+ output = struct ("algorithm", "nnls-pqp", "iterations", iter);
+ endif
+ if (nargout > 4)
+ lambda = zeros (size (x));
+ lambda(p) = w;
+ endif
+
+endfunction
+
+## Tests
+%!test
+%! C = [5 2;2 2];
+%! d = [3; -1];
+%! assert (pqpnonneg (C, d), [0;0.5], 100*eps)
+
+## Test equivalence of lsq and pqp
+%!test
+%! C = rand (20, 10);
+%! d = rand (20, 1);
+%! assert (pqpnonneg (C'*C, -C'*d), lsqnonneg (C, d), 100*eps)