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annotate scripts/optimization/lsqnonneg.m @ 9665:1dba57e9d08d
use blas_trans_type for xgemm
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Sat, 26 Sep 2009 10:41:07 +0200 |
| parents | 36d885c4a1ac |
| children | be55736a0783 |
| rev | line source |
|---|---|
| 7682 | 1 ## Copyright (C) 2008 Bill Denney |
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2 ## Copyright (C) 2008 Jaroslav Hajek |
| 8600 | 3 ## Copyright (C) 2009 VZLU Prague |
| 7682 | 4 ## |
| 5 ## This file is part of Octave. | |
| 6 ## | |
| 7 ## Octave is free software; you can redistribute it and/or modify it | |
| 8 ## under the terms of the GNU General Public License as published by | |
| 9 ## the Free Software Foundation; either version 3 of the License, or (at | |
| 10 ## your option) any later version. | |
| 11 ## | |
| 12 ## Octave is distributed in the hope that it will be useful, but | |
| 13 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 14 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 15 ## General Public License for more details. | |
| 16 ## | |
| 17 ## You should have received a copy of the GNU General Public License | |
| 18 ## along with Octave; see the file COPYING. If not, see | |
| 19 ## <http://www.gnu.org/licenses/>. | |
| 20 | |
| 21 ## -*- texinfo -*- | |
| 22 ## @deftypefn {Function File} {@var{x} =} lsqnonneg (@var{c}, @var{d}) | |
| 23 ## @deftypefnx {Function File} {@var{x} =} lsqnonneg (@var{c}, @var{d}, @var{x0}) | |
| 24 ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}] =} lsqnonneg (@dots{}) | |
| 25 ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}] =} lsqnonneg (@dots{}) | |
| 26 ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}] =} lsqnonneg (@dots{}) | |
| 27 ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}] =} lsqnonneg (@dots{}) | |
| 28 ## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}] =} lsqnonneg (@dots{}) | |
| 29 ## Minimize @code{norm (@var{c}*@var{x}-d)} subject to @code{@var{x} >= | |
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30 ## 0}. @var{c} and @var{d} must be real. @var{x0} is an optional |
| 7682 | 31 ## initial guess for @var{x}. |
| 32 ## | |
| 33 ## Outputs: | |
| 34 ## @itemize @bullet | |
| 35 ## @item resnorm | |
| 36 ## | |
| 37 ## The squared 2-norm of the residual: norm(@var{c}*@var{x}-@var{d})^2 | |
| 38 ## @item residual | |
| 39 ## | |
| 40 ## The residual: @var{d}-@var{c}*@var{x} | |
| 41 ## @item exitflag | |
| 42 ## | |
| 43 ## An indicator of convergence. 0 indicates that the iteration count | |
| 44 ## was exceeded, and therefore convergence was not reached; >0 indicates | |
| 45 ## that the algorithm converged. (The algorithm is stable and will | |
| 46 ## converge given enough iterations.) | |
| 47 ## @item output | |
| 48 ## | |
| 49 ## A structure with two fields: | |
| 50 ## @itemize @bullet | |
| 51 ## @item "algorithm": The algorithm used ("nnls") | |
| 52 ## @item "iterations": The number of iterations taken. | |
| 53 ## @end itemize | |
| 54 ## @item lambda | |
| 55 ## | |
| 56 ## Not implemented. | |
| 57 ## @end itemize | |
| 9635 | 58 ## @seealso{optimset, pqpnonneg} |
| 7682 | 59 ## @end deftypefn |
| 60 | |
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61 ## PKG_ADD: __all_opts__ ("lsqnonneg"); |
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62 |
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63 ## This is implemented from Lawson and Hanson's 1973 algorithm on page |
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64 ## 161 of Solving Least Squares Problems. |
| 7682 | 65 |
| 8600 | 66 function [x, resnorm, residual, exitflag, output, lambda] = lsqnonneg (c, d, x = [], options = struct ()) |
| 67 | |
| 68 if (nargin == 1 && ischar (c) && strcmp (c, 'defaults')) | |
| 69 x = optimset ("MaxIter", 1e5); | |
| 70 return | |
| 71 endif | |
| 72 | |
| 73 if (! (nargin >= 2 && nargin <= 4 && ismatrix (c) && ismatrix (d) && isstruct (options))) | |
| 74 print_usage (); | |
| 75 endif | |
| 7682 | 76 |
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77 ## Lawson-Hanson Step 1 (LH1): initialize the variables. |
| 8600 | 78 m = rows (c); |
| 79 n = columns (c); | |
| 7682 | 80 if (isempty (x)) |
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81 ## Initial guess is 0s. |
| 8600 | 82 x = zeros (n, 1); |
| 83 else | |
| 84 ## ensure nonnegative guess. | |
| 85 x = max (x, 0); | |
| 7682 | 86 endif |
| 87 | |
| 8600 | 88 useqr = m >= n; |
| 8507 | 89 max_iter = optimget (options, "MaxIter", 1e5); |
| 7682 | 90 |
| 8600 | 91 ## Initialize P, according to zero pattern of x. |
| 92 p = find (x > 0).'; | |
| 93 if (useqr) | |
| 94 ## Initialize the QR factorization, economized form. | |
| 95 [q, r] = qr (c(:,p), 0); | |
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96 endif |
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97 |
| 7682 | 98 iter = 0; |
| 8600 | 99 |
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100 ## LH3: test for completion. |
| 8600 | 101 while (iter < max_iter) |
| 102 while (iter < max_iter) | |
| 103 iter++; | |
| 104 | |
| 105 ## LH6: compute the positive matrix and find the min norm solution | |
| 106 ## of the positive problem. | |
| 107 if (useqr) | |
| 108 xtmp = r \ q'*d; | |
| 109 else | |
| 110 xtmp = c(:,p) \ d; | |
| 111 endif | |
| 112 idx = find (xtmp < 0); | |
| 113 | |
| 114 if (isempty (idx)) | |
| 115 ## LH7: tmp solution found, iterate. | |
| 116 x(:) = 0; | |
| 117 x(p) = xtmp; | |
| 118 break; | |
| 119 else | |
| 120 ## LH8, LH9: find the scaling factor. | |
| 121 pidx = p(idx); | |
| 122 sf = x(pidx)./(x(pidx) - xtmp(idx)); | |
| 123 alpha = min (sf); | |
| 124 ## LH10: adjust X. | |
| 125 xx = zeros (n, 1); | |
| 126 xx(p) = xtmp; | |
| 127 x += alpha*(xx - x); | |
| 128 ## LH11: move from P to Z all X == 0. | |
| 129 ## This corresponds to those indices where minimum of sf is attained. | |
| 130 idx = idx (sf == alpha); | |
| 131 p(idx) = []; | |
| 132 if (useqr) | |
| 133 ## update the QR factorization. | |
| 134 [q, r] = qrdelete (q, r, idx); | |
| 135 endif | |
| 136 endif | |
| 137 endwhile | |
| 138 | |
| 139 ## compute the gradient. | |
| 140 w = c'*(d - c*x); | |
| 141 w(p) = []; | |
| 142 if (! any (w > 0)) | |
| 143 if (useqr) | |
| 144 ## verify the solution achieved using qr updating. | |
| 145 ## in the best case, this should only take a single step. | |
| 146 useqr = false; | |
| 147 continue; | |
| 148 else | |
| 149 ## we're finished. | |
| 150 break; | |
| 151 endif | |
| 152 endif | |
| 153 | |
| 154 ## find the maximum gradient. | |
| 7682 | 155 idx = find (w == max (w)); |
| 156 if (numel (idx) > 1) | |
| 157 warning ("lsqnonneg:nonunique", | |
| 158 "A non-unique solution may be returned due to equal gradients."); | |
| 159 idx = idx(1); | |
| 160 endif | |
| 8600 | 161 ## move the index from Z to P. Keep P sorted. |
| 162 z = [1:n]; z(p) = []; | |
| 163 zidx = z(idx); | |
| 164 jdx = 1 + lookup (p, zidx); | |
| 165 p = [p(1:jdx-1), zidx, p(jdx:end)]; | |
| 166 if (useqr) | |
| 167 ## insert the column into the QR factorization. | |
| 168 [q, r] = qrinsert (q, r, jdx, c(:,zidx)); | |
| 169 endif | |
| 7682 | 170 |
| 171 endwhile | |
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172 ## LH12: complete. |
| 7682 | 173 |
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174 ## Generate the additional output arguments. |
| 7682 | 175 if (nargout > 1) |
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176 resnorm = norm (c*x - d) ^ 2; |
| 7682 | 177 endif |
| 178 if (nargout > 2) | |
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parents:
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179 residual = d - c*x; |
| 7682 | 180 endif |
| 181 exitflag = iter; | |
| 8507 | 182 if (nargout > 3 && iter >= max_iter) |
| 7682 | 183 exitflag = 0; |
| 184 endif | |
| 185 if (nargout > 4) | |
| 186 output = struct ("algorithm", "nnls", "iterations", iter); | |
| 187 endif | |
| 188 if (nargout > 5) | |
| 8600 | 189 lambda = zeros (size (x)); |
| 190 lambda(p) = w; | |
| 7682 | 191 endif |
| 192 | |
| 193 endfunction | |
| 194 | |
| 195 ## Tests | |
| 196 %!test | |
| 197 %! C = [1 0;0 1;2 1]; | |
| 198 %! d = [1;3;-2]; | |
| 199 %! assert (lsqnonneg (C, d), [0;0.5], 100*eps) | |
| 200 | |
| 201 %!test | |
| 202 %! C = [0.0372 0.2869;0.6861 0.7071;0.6233 0.6245;0.6344 0.6170]; | |
| 203 %! d = [0.8587;0.1781;0.0747;0.8405]; | |
| 204 %! xnew = [0;0.6929]; | |
| 205 %! assert (lsqnonneg (C, d), xnew, 0.0001) |