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view scripts/optimization/lsqnonneg.m @ 8507:cadc73247d65
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author | John W. Eaton <jwe@octave.org> |
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date | Tue, 13 Jan 2009 14:08:36 -0500 |
parents | 096c22ce2b0b |
children | a6c1aa6f5915 |
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## Copyright (C) 2008 Bill Denney ## Copyright (C) 2008 Jaroslav Hajek ## ## 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} =} lsqnonneg (@var{c}, @var{d}) ## @deftypefnx {Function File} {@var{x} =} lsqnonneg (@var{c}, @var{d}, @var{x0}) ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}] =} lsqnonneg (@dots{}) ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}] =} lsqnonneg (@dots{}) ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}] =} lsqnonneg (@dots{}) ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}] =} lsqnonneg (@dots{}) ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}] =} lsqnonneg (@dots{}) ## Minimize @code{norm (@var{c}*@var{x}-d)} subject to @code{@var{x} >= ## 0}. @var{c} and @var{d} must be real. @var{x0} is an optional ## initial guess for @var{x}. ## ## Outputs: ## @itemize @bullet ## @item resnorm ## ## The squared 2-norm of the residual: norm(@var{c}*@var{x}-@var{d})^2 ## @item residual ## ## The residual: @var{d}-@var{c}*@var{x} ## @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} ## @end deftypefn ## This is implemented from Lawson and Hanson's 1973 algorithm on page ## 161 of Solving Least Squares Problems. function [x, resnorm, residual, exitflag, output, lambda] = lsqnonneg (c, d, x = [], options = []) ## Lawson-Hanson Step 1 (LH1): initialize the variables. if (isempty (x)) ## Initial guess is 0s. x = zeros (columns (c), 1); endif max_iter = optimget (options, "MaxIter", 1e5); ## Initialize the values. p = false (1, numel (x)); z = !p; ## If the problem is significantly over-determined, preprocess it using a ## QR factorization first. if (rows (c) >= 1.5 * columns (c)) [q, r] = qr (c, 0); d = q'*d; c = r; clear q endif ## LH2: compute the gradient. w = c'*(d - c*x); xtmp = zeros (columns (c), 1); iter = 0; ## LH3: test for completion. while (any (z) && any (w(z) > 0) && iter < max_iter) ## LH4: find the maximum gradient. idx = find (w == max (w)); if (numel (idx) > 1) warning ("lsqnonneg:nonunique", "A non-unique solution may be returned due to equal gradients."); idx = idx(1); endif ## LH5: move the index from Z to P. z(idx) = false; p(idx) = true; newx = false; while (! newx && iter < max_iter) iter++; ## LH6: compute the positive matrix and find the min norm solution ## of the positive problem. ## Find min norm solution to the positive matrix. xtmp(:) = 0; xtmp(p) = c(:,p) \ d; if (all (xtmp >= 0)) ## LH7: tmp solution found, iterate. newx = true; x = xtmp; else ## LH8, LH9: find the scaling factor and adjust X. mask = (xtmp < 0); alpha = min (x(mask)./(x(mask) - xtmp(mask))); ## LH10: adjust X. x = x + alpha*(xtmp - x); ## LH11: move from P to Z all X == 0. z |= (x == 0); p = !z; endif endwhile w = c'*(d - c*x); endwhile ## LH12: complete. ## Generate the additional output arguments. if (nargout > 1) resnorm = norm (c*x - d) ^ 2; endif if (nargout > 2) residual = d - c*x; endif exitflag = iter; if (nargout > 3 && iter >= max_iter) exitflag = 0; endif if (nargout > 4) output = struct ("algorithm", "nnls", "iterations", iter); endif if (nargout > 5) lambda = w; endif endfunction ## Tests %!test %! C = [1 0;0 1;2 1]; %! d = [1;3;-2]; %! assert (lsqnonneg (C, d), [0;0.5], 100*eps) %!test %! C = [0.0372 0.2869;0.6861 0.7071;0.6233 0.6245;0.6344 0.6170]; %! d = [0.8587;0.1781;0.0747;0.8405]; %! xnew = [0;0.6929]; %! assert (lsqnonneg (C, d), xnew, 0.0001)