Mercurial > octave-nkf
annotate scripts/optimization/qp.m @ 10793:be55736a0783
Grammarcheck the documentation from m-files.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Sun, 18 Jul 2010 20:35:16 -0700 |
| parents | 95c3e38098bf |
| children | 693e22af08ae |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2000, 2001, 2004, 2005, 2006, 2007, 2008, |
| 2 ## 2009 Gabriele Pannocchia. | |
| 5289 | 3 ## |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
| 9 ## your option) any later version. | |
| 5289 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 7016 | 17 ## along with Octave; see the file COPYING. If not, see |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 5289 | 19 |
| 20 ## -*- texinfo -*- | |
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21 ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}) |
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22 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}) |
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23 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}) |
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24 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub}) |
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25 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, @var{A_in}, @var{A_ub}) |
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26 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@dots{}, @var{options}) |
| 5289 | 27 ## Solve the quadratic program |
| 6741 | 28 ## @tex |
| 29 ## $$ | |
| 30 ## \min_x {1 \over 2} x^T H x + x^T q | |
| 31 ## $$ | |
| 32 ## @end tex | |
| 33 ## @ifnottex | |
| 5289 | 34 ## |
| 35 ## @example | |
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36 ## @group |
| 5289 | 37 ## min 0.5 x'*H*x + x'*q |
| 38 ## x | |
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39 ## @end group |
| 5289 | 40 ## @end example |
| 41 ## | |
| 6741 | 42 ## @end ifnottex |
| 43 ## subject to | |
| 5289 | 44 ## @tex |
| 6741 | 45 ## $$ |
| 46 ## Ax = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} \leq A_{ub} | |
| 47 ## $$ | |
| 5289 | 48 ## @end tex |
| 6741 | 49 ## @ifnottex |
| 5289 | 50 ## |
| 51 ## @example | |
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52 ## @group |
| 5289 | 53 ## A*x = b |
| 54 ## lb <= x <= ub | |
| 6741 | 55 ## A_lb <= A_in*x <= A_ub |
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56 ## @end group |
| 5289 | 57 ## @end example |
| 6741 | 58 ## @end ifnottex |
| 5289 | 59 ## |
| 60 ## @noindent | |
| 61 ## using a null-space active-set method. | |
| 62 ## | |
| 63 ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, | |
| 64 ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not | |
| 65 ## present. If the initial guess is feasible the algorithm is faster. | |
| 66 ## | |
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67 ## @table @var |
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68 ## @item options |
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69 ## An optional structure containing the following |
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70 ## parameter(s) used to define the behavior of the solver. Missing elements |
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71 ## in the structure take on default values, so you only need to set the |
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72 ## elements that you wish to change from the default. |
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73 ## |
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74 ## @table @code |
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75 ## @item MaxIter (default: 200) |
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76 ## Maximum number of iterations. |
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77 ## @end table |
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78 ## @end table |
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79 ## |
| 5289 | 80 ## The value @var{info} is a structure with the following fields: |
| 81 ## @table @code | |
| 82 ## @item solveiter | |
| 83 ## The number of iterations required to find the solution. | |
| 84 ## @item info | |
| 85 ## An integer indicating the status of the solution, as follows: | |
| 86 ## @table @asis | |
| 87 ## @item 0 | |
| 88 ## The problem is feasible and convex. Global solution found. | |
| 89 ## @item 1 | |
| 90 ## The problem is not convex. Local solution found. | |
| 91 ## @item 2 | |
| 92 ## The problem is not convex and unbounded. | |
| 93 ## @item 3 | |
| 94 ## Maximum number of iterations reached. | |
| 95 ## @item 6 | |
| 96 ## The problem is infeasible. | |
| 97 ## @end table | |
| 98 ## @end table | |
| 99 ## @end deftypefn | |
| 100 | |
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101 ## PKG_ADD: __all_opts__ ("qp"); |
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102 |
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103 function [x, obj, INFO, lambda] = qp (x0, H, varargin) |
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104 |
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105 nargs = nargin; |
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106 |
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107 if (nargin == 1 && ischar (x0) && strcmp (x0, 'defaults')) |
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108 x = optimset ("MaxIter", 200); |
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109 return; |
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110 endif |
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111 |
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112 if (nargs > 2 && isstruct (varargin{end})) |
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113 options = varargin{end}; |
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114 nargs--; |
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115 else |
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116 options = struct (); |
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117 endif |
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118 |
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119 if (nargs >= 3) |
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120 q = varargin{1}; |
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121 else |
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122 q = []; |
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123 endif |
| 5289 | 124 |
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125 if (nargs >= 5) |
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126 A = varargin{2}; |
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127 b = varargin{3}; |
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128 else |
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129 A = []; |
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130 b = []; |
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131 endif |
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132 |
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133 if (nargs >= 7) |
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134 lb = varargin{4}; |
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135 ub = varargin{5}; |
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136 else |
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137 lb = []; |
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138 ub = []; |
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139 endif |
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140 |
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141 if (nargs == 10) |
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142 A_lb = varargin{6}; |
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143 A_in = varargin{7}; |
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144 A_ub = varargin{8}; |
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145 else |
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146 A_lb = []; |
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147 A_in = []; |
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148 A_ub = []; |
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149 endif |
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150 |
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151 if (nargs == 2 || nargs == 3 || nargs == 5 || nargs == 7 || nargs == 10) |
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152 |
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153 maxit = optimget (options, "MaxIter", 200); |
| 5289 | 154 |
| 155 ## Checking the quadratic penalty | |
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156 if (! issquare (H)) |
| 5289 | 157 error ("qp: quadratic penalty matrix not square"); |
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158 elseif (! ishermitian (H)) |
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159 ## warning ("qp: quadratic penalty matrix not hermitian"); |
| 5289 | 160 H = (H + H')/2; |
| 161 endif | |
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162 n = rows (H); |
| 5289 | 163 |
| 164 ## Checking the initial guess (if empty it is resized to the | |
| 165 ## right dimension and filled with 0) | |
| 166 if (isempty (x0)) | |
| 167 x0 = zeros (n, 1); | |
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168 elseif (numel (x0) != n) |
| 5289 | 169 error ("qp: the initial guess has incorrect length"); |
| 170 endif | |
| 171 | |
| 172 ## Linear penalty. | |
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173 if (isempty (q)) |
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174 q = zeros (n, 1); |
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175 elseif (numel (q) != n) |
| 5289 | 176 error ("qp: the linear term has incorrect length"); |
| 177 endif | |
| 178 | |
| 179 ## Equality constraint matrices | |
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180 if (isempty (A) || isempty (b)) |
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181 A = zeros (0, n); |
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182 b = zeros (0, 1); |
| 5289 | 183 n_eq = 0; |
| 184 else | |
| 185 [n_eq, n1] = size (A); | |
| 186 if (n1 != n) | |
| 10549 | 187 error ("qp: equality constraint matrix has incorrect column dimension"); |
| 5289 | 188 endif |
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189 if (numel (b) != n_eq) |
| 10549 | 190 error ("qp: equality constraint matrix and vector have inconsistent dimension"); |
| 5289 | 191 endif |
| 192 endif | |
| 193 | |
| 194 ## Bound constraints | |
| 195 Ain = zeros (0, n); | |
| 196 bin = zeros (0, 1); | |
| 197 n_in = 0; | |
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198 if (nargs > 5) |
| 5289 | 199 if (! isempty (lb)) |
| 10549 | 200 if (numel (lb) != n) |
| 201 error ("qp: lower bound has incorrect length"); | |
| 202 elseif (isempty (ub)) | |
| 203 Ain = [Ain; eye(n)]; | |
| 204 bin = [bin; lb]; | |
| 205 endif | |
| 5289 | 206 endif |
| 207 | |
| 208 if (! isempty (ub)) | |
| 10549 | 209 if (numel (ub) != n) |
| 210 error ("qp: upper bound has incorrect length"); | |
| 211 elseif (isempty (lb)) | |
| 212 Ain = [Ain; -eye(n)]; | |
| 213 bin = [bin; -ub]; | |
| 214 endif | |
| 5289 | 215 endif |
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216 |
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217 if (! isempty (lb) && ! isempty (ub)) |
| 10549 | 218 rtol = sqrt (eps); |
| 219 for i = 1:n | |
| 220 if (abs(lb (i) - ub(i)) < rtol*(1 + max (abs (lb(i) + ub(i))))) | |
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221 ## These are actually an equality constraint |
| 10549 | 222 tmprow = zeros(1,n); |
| 223 tmprow(i) = 1; | |
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224 A = [A;tmprow]; |
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225 b = [b; 0.5*(lb(i) + ub(i))]; |
| 10549 | 226 n_eq = n_eq + 1; |
| 227 else | |
| 228 tmprow = zeros(1,n); | |
| 229 tmprow(i) = 1; | |
| 230 Ain = [Ain; tmprow; -tmprow]; | |
| 231 bin = [bin; lb(i); -ub(i)]; | |
| 232 n_in = n_in + 2; | |
| 233 endif | |
| 234 endfor | |
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235 endif |
| 5289 | 236 endif |
| 237 | |
| 238 ## Inequality constraints | |
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239 if (nargs > 7) |
| 5289 | 240 [dimA_in, n1] = size (A_in); |
| 241 if (n1 != n) | |
| 10549 | 242 error ("qp: inequality constraint matrix has incorrect column dimension"); |
| 5289 | 243 else |
| 10549 | 244 if (! isempty (A_lb)) |
| 245 if (numel (A_lb) != dimA_in) | |
| 246 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); | |
| 247 elseif (isempty (A_ub)) | |
| 248 Ain = [Ain; A_in]; | |
| 249 bin = [bin; A_lb]; | |
| 250 endif | |
| 251 endif | |
| 252 if (! isempty (A_ub)) | |
| 253 if (numel (A_ub) != dimA_in) | |
| 254 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); | |
| 255 elseif (isempty (A_lb)) | |
| 256 Ain = [Ain; -A_in]; | |
| 257 bin = [bin; -A_ub]; | |
| 258 endif | |
| 259 endif | |
| 260 | |
| 261 if (! isempty (A_lb) && ! isempty (A_ub)) | |
| 262 rtol = sqrt (eps); | |
| 263 for i = 1:dimA_in | |
| 264 if (abs (A_lb(i) - A_ub(i)) < rtol*(1 + max (abs (A_lb(i) + A_ub(i))))) | |
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265 ## These are actually an equality constraint |
| 10549 | 266 tmprow = A_in(i,:); |
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267 A = [A;tmprow]; |
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268 b = [b; 0.5*(A_lb(i) + A_ub(i))]; |
| 10549 | 269 n_eq = n_eq + 1; |
| 270 else | |
| 271 tmprow = A_in(i,:); | |
| 272 Ain = [Ain; tmprow; -tmprow]; | |
| 273 bin = [bin; A_lb(i); -A_ub(i)]; | |
| 274 n_in = n_in + 2; | |
| 275 endif | |
| 276 endfor | |
| 277 endif | |
| 5289 | 278 endif |
| 279 endif | |
| 280 | |
| 281 ## Now we should have the following QP: | |
| 282 ## | |
| 283 ## min_x 0.5*x'*H*x + x'*q | |
| 284 ## s.t. A*x = b | |
| 285 ## Ain*x >= bin | |
| 286 | |
| 287 ## Discard inequality constraints that have -Inf bounds since those | |
| 288 ## will never be active. | |
| 289 idx = isinf (bin) & bin < 0; | |
| 6523 | 290 |
| 6526 | 291 bin(idx) = []; |
| 292 Ain(idx,:) = []; | |
| 5289 | 293 |
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294 n_in = numel (bin); |
| 5310 | 295 |
| 5289 | 296 ## Check if the initial guess is feasible. |
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297 if (isa (x0, "single") || isa (H, "single") || isa (q, "single") || isa (A, "single") |
| 10549 | 298 || isa (b, "single")) |
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299 rtol = sqrt (eps ("single")); |
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300 else |
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301 rtol = sqrt (eps); |
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302 endif |
| 5289 | 303 |
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304 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+abs (b))); |
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305 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+abs (bin)))); |
| 5289 | 306 |
| 307 info = 0; | |
| 308 if (eq_infeasible || in_infeasible) | |
| 309 ## The initial guess is not feasible. | |
| 310 ## First define xbar that is feasible with respect to the equality | |
| 311 ## constraints. | |
| 312 if (eq_infeasible) | |
| 10549 | 313 if (rank (A) < n_eq) |
| 314 error ("qp: equality constraint matrix must be full row rank") | |
| 315 endif | |
| 316 xbar = pinv (A) * b; | |
| 5289 | 317 else |
| 10549 | 318 xbar = x0; |
| 5289 | 319 endif |
| 320 | |
| 321 ## Check if xbar is feasible with respect to the inequality | |
| 322 ## constraints also. | |
| 323 if (n_in > 0) | |
| 10549 | 324 res = Ain * xbar - bin; |
| 325 if (any (res < -rtol * (1 + abs (bin)))) | |
| 326 ## xbar is not feasible with respect to the inequality | |
| 327 ## constraints. Compute a step in the null space of the | |
| 328 ## equality constraints, by solving a QP. If the slack is | |
| 329 ## small, we have a feasible initial guess. Otherwise, the | |
| 330 ## problem is infeasible. | |
| 331 if (n_eq > 0) | |
| 332 Z = null (A); | |
| 333 if (isempty (Z)) | |
| 334 ## The problem is infeasible because A is square and full | |
| 335 ## rank, but xbar is not feasible. | |
| 336 info = 6; | |
| 337 endif | |
| 338 endif | |
| 5289 | 339 |
| 10549 | 340 if (info != 6) |
| 5289 | 341 ## Solve an LP with additional slack variables to find |
| 10549 | 342 ## a feasible starting point. |
| 343 gamma = eye (n_in); | |
| 344 if (n_eq > 0) | |
| 345 Atmp = [Ain*Z, gamma]; | |
| 346 btmp = -res; | |
| 347 else | |
| 348 Atmp = [Ain, gamma]; | |
| 349 btmp = bin; | |
| 350 endif | |
| 351 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; | |
| 352 lb = [-Inf(n-n_eq,1); zeros(n_in,1)]; | |
| 353 ub = []; | |
| 354 ctype = repmat ("L", n_in, 1); | |
| 355 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); | |
| 356 if ((status == 180 || status == 181 || status == 151) | |
| 357 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) | |
| 358 ## We found a feasible starting point | |
| 359 if (n_eq > 0) | |
| 360 x0 = xbar + Z*P(1:n-n_eq); | |
| 361 else | |
| 362 x0 = P(1:n); | |
| 5289 | 363 endif |
| 10549 | 364 else |
| 365 ## The problem is infeasible | |
| 366 info = 6; | |
| 367 endif | |
| 368 endif | |
| 369 else | |
| 370 ## xbar is feasible. We use it a starting point. | |
| 371 x0 = xbar; | |
| 372 endif | |
| 5289 | 373 else |
| 10549 | 374 ## xbar is feasible. We use it a starting point. |
| 375 x0 = xbar; | |
| 5289 | 376 endif |
| 377 endif | |
| 378 | |
| 379 if (info == 0) | |
| 380 ## The initial (or computed) guess is feasible. | |
| 381 ## We call the solver. | |
| 382 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); | |
| 383 else | |
| 384 iter = 0; | |
| 385 x = x0; | |
| 386 lambda = []; | |
| 387 endif | |
| 388 obj = 0.5 * x' * H * x + q' * x; | |
| 389 INFO.solveiter = iter; | |
| 390 INFO.info = info; | |
| 391 | |
| 392 else | |
| 6046 | 393 print_usage (); |
| 5289 | 394 endif |
| 395 | |
| 396 endfunction |