Mercurial > octave
annotate scripts/optimization/pqpnonneg.m @ 14138:72c96de7a403 stable
maint: update copyright notices for 2012
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Mon, 02 Jan 2012 14:25:41 -0500 |
| parents | b9a89ca0fb75 |
| children | f3d52523cde1 |
| rev | line source |
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maint: update copyright notices for 2012
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1 ## Copyright (C) 2008-2012 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 | |
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61 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. |
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62 ## PKG_ADD: [~] = __all_opts__ ("pqpnonneg"); |
| 9635 | 63 |
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64 ## This is analogical to the lsqnonneg implementation, which is |
| 9635 | 65 ## implemented from Lawson and Hanson's 1973 algorithm on page |
| 66 ## 161 of Solving Least Squares Problems. | |
| 67 ## It shares the convergence guarantees. | |
| 68 | |
| 69 function [x, minval, exitflag, output, lambda] = pqpnonneg (c, d, x = [], options = struct ()) | |
| 70 | |
| 71 if (nargin == 1 && ischar (c) && strcmp (c, 'defaults')) | |
| 72 x = optimset ("MaxIter", 1e5); | |
| 73 return | |
| 74 endif | |
| 75 | |
| 76 if (! (nargin >= 2 && nargin <= 4 && ismatrix (c) && ismatrix (d) && isstruct (options))) | |
| 77 print_usage (); | |
| 78 endif | |
| 79 | |
| 80 ## Lawson-Hanson Step 1 (LH1): initialize the variables. | |
| 81 m = rows (c); | |
| 82 n = columns (c); | |
| 83 if (m != n) | |
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84 error ("pqpnonneg: matrix must be square"); |
| 9635 | 85 endif |
| 86 | |
| 87 if (isempty (x)) | |
| 88 ## Initial guess is 0s. | |
| 89 x = zeros (n, 1); | |
| 90 else | |
| 91 ## ensure nonnegative guess. | |
| 92 x = max (x, 0); | |
| 93 endif | |
| 94 | |
| 95 max_iter = optimget (options, "MaxIter", 1e5); | |
| 96 | |
| 97 ## Initialize P, according to zero pattern of x. | |
| 98 p = find (x > 0).'; | |
| 99 ## Initialize the Cholesky factorization. | |
| 100 r = chol (c(p, p)); | |
| 101 usechol = true; | |
| 102 | |
| 103 iter = 0; | |
| 104 | |
| 105 ## LH3: test for completion. | |
| 106 while (iter < max_iter) | |
| 107 while (iter < max_iter) | |
| 108 iter++; | |
| 109 | |
| 110 ## LH6: compute the positive matrix and find the min norm solution | |
| 111 ## of the positive problem. | |
| 112 if (usechol) | |
| 113 xtmp = -(r \ (r' \ d(p))); | |
| 114 else | |
| 115 xtmp = -(c(p,p) \ d(p)); | |
| 116 endif | |
| 117 idx = find (xtmp < 0); | |
| 118 | |
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119 if (isempty (idx)) |
| 9635 | 120 ## LH7: tmp solution found, iterate. |
| 121 x(:) = 0; | |
| 122 x(p) = xtmp; | |
| 123 break; | |
| 124 else | |
| 125 ## LH8, LH9: find the scaling factor. | |
| 126 pidx = p(idx); | |
| 127 sf = x(pidx)./(x(pidx) - xtmp(idx)); | |
| 128 alpha = min (sf); | |
| 129 ## LH10: adjust X. | |
| 130 xx = zeros (n, 1); | |
| 131 xx(p) = xtmp; | |
| 132 x += alpha*(xx - x); | |
| 133 ## LH11: move from P to Z all X == 0. | |
| 134 ## This corresponds to those indices where minimum of sf is attained. | |
| 135 idx = idx (sf == alpha); | |
| 136 p(idx) = []; | |
| 137 if (usechol) | |
| 138 ## update the Cholesky factorization. | |
| 139 r = choldelete (r, idx); | |
| 140 endif | |
| 141 endif | |
| 142 endwhile | |
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143 |
| 9635 | 144 ## compute the gradient. |
| 145 w = -(d + c*x); | |
| 146 w(p) = []; | |
| 147 if (! any (w > 0)) | |
| 148 if (usechol) | |
| 149 ## verify the solution achieved using qr updating. | |
| 150 ## in the best case, this should only take a single step. | |
| 151 usechol = false; | |
| 152 continue; | |
| 153 else | |
| 154 ## we're finished. | |
| 155 break; | |
| 156 endif | |
| 157 endif | |
| 158 | |
| 159 ## find the maximum gradient. | |
| 160 idx = find (w == max (w)); | |
| 161 if (numel (idx) > 1) | |
| 162 warning ("pqpnonneg:nonunique", | |
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163 "a non-unique solution may be returned due to equal gradients"); |
| 9635 | 164 idx = idx(1); |
| 165 endif | |
| 166 ## move the index from Z to P. Keep P sorted. | |
| 167 z = [1:n]; z(p) = []; | |
| 168 zidx = z(idx); | |
| 169 jdx = 1 + lookup (p, zidx); | |
| 170 p = [p(1:jdx-1), zidx, p(jdx:end)]; | |
| 171 if (usechol) | |
| 172 ## insert the column into the Cholesky factorization. | |
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173 [r, bad] = cholinsert (r, jdx, c(p,zidx)); |
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174 if (bad) |
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175 ## If the insertion failed, we switch off updates and go on. |
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176 usechol = false; |
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177 endif |
| 9635 | 178 endif |
| 179 | |
| 180 endwhile | |
| 181 ## LH12: complete. | |
| 182 | |
| 183 ## Generate the additional output arguments. | |
| 184 if (nargout > 1) | |
| 185 minval = 1/2*(x'*c*x) + d'*x; | |
| 186 endif | |
| 187 exitflag = iter; | |
| 188 if (nargout > 2 && iter >= max_iter) | |
| 189 exitflag = 0; | |
| 190 endif | |
| 191 if (nargout > 3) | |
| 192 output = struct ("algorithm", "nnls-pqp", "iterations", iter); | |
| 193 endif | |
| 194 if (nargout > 4) | |
| 195 lambda = zeros (size (x)); | |
| 196 lambda(p) = w; | |
| 197 endif | |
| 198 | |
| 199 endfunction | |
| 200 | |
| 201 ## Tests | |
| 202 %!test | |
| 203 %! C = [5 2;2 2]; | |
| 204 %! d = [3; -1]; | |
| 205 %! assert (pqpnonneg (C, d), [0;0.5], 100*eps) | |
| 206 | |
| 207 ## Test equivalence of lsq and pqp | |
| 208 %!test | |
| 209 %! C = rand (20, 10); | |
| 210 %! d = rand (20, 1); | |
| 211 %! assert (pqpnonneg (C'*C, -C'*d), lsqnonneg (C, d), 100*eps) |