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annotate scripts/optimization/pqpnonneg.m @ 11523:fd0a3ac60b0e
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Fri, 14 Jan 2011 05:47:45 -0500 |
| parents | 693e22af08ae |
| children | c792872f8942 |
| rev | line source |
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| 11523 | 1 ## Copyright (C) 2008-2011 Bill Denney |
| 9635 | 2 ## Copyright (C) 2008 Jaroslav Hajek |
| 3 ## Copyright (C) 2009 VZLU Prague | |
| 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 -*- | |
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22 ## @deftypefn {Function File} {@var{x} =} pqpnonneg (@var{c}, @var{d}) |
| 9635 | 23 ## @deftypefnx {Function File} {@var{x} =} pqpnonneg (@var{c}, @var{d}, @var{x0}) |
| 24 ## @deftypefnx {Function File} {[@var{x}, @var{minval}] =} pqpnonneg (@dots{}) | |
| 25 ## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}] =} pqpnonneg (@dots{}) | |
| 26 ## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}] =} pqpnonneg (@dots{}) | |
| 27 ## @deftypefnx {Function File} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}, @var{lambda}] =} pqpnonneg (@dots{}) | |
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28 ## Minimize @code{1/2*x'*c*x + d'*x} subject to @code{@var{x} >= 0}. @var{c} |
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29 ## and @var{d} must be real, and @var{c} must be symmetric and positive |
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30 ## definite. @var{x0} is an optional initial guess for @var{x}. |
| 9635 | 31 ## |
| 32 ## Outputs: | |
| 33 ## @itemize @bullet | |
| 34 ## @item minval | |
| 35 ## | |
| 36 ## The minimum attained model value, 1/2*xmin'*c*xmin + d'*xmin | |
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37 ## |
| 9635 | 38 ## @item exitflag |
| 39 ## | |
| 40 ## An indicator of convergence. 0 indicates that the iteration count | |
| 41 ## was exceeded, and therefore convergence was not reached; >0 indicates | |
| 42 ## that the algorithm converged. (The algorithm is stable and will | |
| 43 ## converge given enough iterations.) | |
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44 ## |
| 9635 | 45 ## @item output |
| 46 ## | |
| 47 ## A structure with two fields: | |
| 48 ## @itemize @bullet | |
| 49 ## @item "algorithm": The algorithm used ("nnls") | |
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50 ## |
| 9635 | 51 ## @item "iterations": The number of iterations taken. |
| 52 ## @end itemize | |
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53 ## |
| 9635 | 54 ## @item lambda |
| 55 ## | |
| 56 ## Not implemented. | |
| 57 ## @end itemize | |
| 58 ## @seealso{optimset, lsqnonneg, qp} | |
| 59 ## @end deftypefn | |
| 60 | |
| 61 ## PKG_ADD: __all_opts__ ("pqpnonneg"); | |
| 62 | |
| 63 ## This is analogical to the lsqnonneg implementation, which is | |
| 64 ## implemented from Lawson and Hanson's 1973 algorithm on page | |
| 65 ## 161 of Solving Least Squares Problems. | |
| 66 ## It shares the convergence guarantees. | |
| 67 | |
| 68 function [x, minval, exitflag, output, lambda] = pqpnonneg (c, d, x = [], options = struct ()) | |
| 69 | |
| 70 if (nargin == 1 && ischar (c) && strcmp (c, 'defaults')) | |
| 71 x = optimset ("MaxIter", 1e5); | |
| 72 return | |
| 73 endif | |
| 74 | |
| 75 if (! (nargin >= 2 && nargin <= 4 && ismatrix (c) && ismatrix (d) && isstruct (options))) | |
| 76 print_usage (); | |
| 77 endif | |
| 78 | |
| 79 ## Lawson-Hanson Step 1 (LH1): initialize the variables. | |
| 80 m = rows (c); | |
| 81 n = columns (c); | |
| 82 if (m != n) | |
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83 error ("pqpnonneg: matrix must be square"); |
| 9635 | 84 endif |
| 85 | |
| 86 if (isempty (x)) | |
| 87 ## Initial guess is 0s. | |
| 88 x = zeros (n, 1); | |
| 89 else | |
| 90 ## ensure nonnegative guess. | |
| 91 x = max (x, 0); | |
| 92 endif | |
| 93 | |
| 94 max_iter = optimget (options, "MaxIter", 1e5); | |
| 95 | |
| 96 ## Initialize P, according to zero pattern of x. | |
| 97 p = find (x > 0).'; | |
| 98 ## Initialize the Cholesky factorization. | |
| 99 r = chol (c(p, p)); | |
| 100 usechol = true; | |
| 101 | |
| 102 iter = 0; | |
| 103 | |
| 104 ## LH3: test for completion. | |
| 105 while (iter < max_iter) | |
| 106 while (iter < max_iter) | |
| 107 iter++; | |
| 108 | |
| 109 ## LH6: compute the positive matrix and find the min norm solution | |
| 110 ## of the positive problem. | |
| 111 if (usechol) | |
| 112 xtmp = -(r \ (r' \ d(p))); | |
| 113 else | |
| 114 xtmp = -(c(p,p) \ d(p)); | |
| 115 endif | |
| 116 idx = find (xtmp < 0); | |
| 117 | |
| 118 if (isempty (idx)) | |
| 119 ## LH7: tmp solution found, iterate. | |
| 120 x(:) = 0; | |
| 121 x(p) = xtmp; | |
| 122 break; | |
| 123 else | |
| 124 ## LH8, LH9: find the scaling factor. | |
| 125 pidx = p(idx); | |
| 126 sf = x(pidx)./(x(pidx) - xtmp(idx)); | |
| 127 alpha = min (sf); | |
| 128 ## LH10: adjust X. | |
| 129 xx = zeros (n, 1); | |
| 130 xx(p) = xtmp; | |
| 131 x += alpha*(xx - x); | |
| 132 ## LH11: move from P to Z all X == 0. | |
| 133 ## This corresponds to those indices where minimum of sf is attained. | |
| 134 idx = idx (sf == alpha); | |
| 135 p(idx) = []; | |
| 136 if (usechol) | |
| 137 ## update the Cholesky factorization. | |
| 138 r = choldelete (r, idx); | |
| 139 endif | |
| 140 endif | |
| 141 endwhile | |
| 142 | |
| 143 ## compute the gradient. | |
| 144 w = -(d + c*x); | |
| 145 w(p) = []; | |
| 146 if (! any (w > 0)) | |
| 147 if (usechol) | |
| 148 ## verify the solution achieved using qr updating. | |
| 149 ## in the best case, this should only take a single step. | |
| 150 usechol = false; | |
| 151 continue; | |
| 152 else | |
| 153 ## we're finished. | |
| 154 break; | |
| 155 endif | |
| 156 endif | |
| 157 | |
| 158 ## find the maximum gradient. | |
| 159 idx = find (w == max (w)); | |
| 160 if (numel (idx) > 1) | |
| 161 warning ("pqpnonneg:nonunique", | |
| 162 "A non-unique solution may be returned due to equal gradients."); | |
| 163 idx = idx(1); | |
| 164 endif | |
| 165 ## move the index from Z to P. Keep P sorted. | |
| 166 z = [1:n]; z(p) = []; | |
| 167 zidx = z(idx); | |
| 168 jdx = 1 + lookup (p, zidx); | |
| 169 p = [p(1:jdx-1), zidx, p(jdx:end)]; | |
| 170 if (usechol) | |
| 171 ## insert the column into the Cholesky factorization. | |
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172 [r, bad] = cholinsert (r, jdx, c(p,zidx)); |
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173 if (bad) |
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174 ## If the insertion failed, we switch off updates and go on. |
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175 usechol = false; |
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176 endif |
| 9635 | 177 endif |
| 178 | |
| 179 endwhile | |
| 180 ## LH12: complete. | |
| 181 | |
| 182 ## Generate the additional output arguments. | |
| 183 if (nargout > 1) | |
| 184 minval = 1/2*(x'*c*x) + d'*x; | |
| 185 endif | |
| 186 exitflag = iter; | |
| 187 if (nargout > 2 && iter >= max_iter) | |
| 188 exitflag = 0; | |
| 189 endif | |
| 190 if (nargout > 3) | |
| 191 output = struct ("algorithm", "nnls-pqp", "iterations", iter); | |
| 192 endif | |
| 193 if (nargout > 4) | |
| 194 lambda = zeros (size (x)); | |
| 195 lambda(p) = w; | |
| 196 endif | |
| 197 | |
| 198 endfunction | |
| 199 | |
| 200 ## Tests | |
| 201 %!test | |
| 202 %! C = [5 2;2 2]; | |
| 203 %! d = [3; -1]; | |
| 204 %! assert (pqpnonneg (C, d), [0;0.5], 100*eps) | |
| 205 | |
| 206 ## Test equivalence of lsq and pqp | |
| 207 %!test | |
| 208 %! C = rand (20, 10); | |
| 209 %! d = rand (20, 1); | |
| 210 %! assert (pqpnonneg (C'*C, -C'*d), lsqnonneg (C, d), 100*eps) |