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annotate scripts/optimization/qp.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 |
|---|---|
| 11523 | 1 ## Copyright (C) 2000-2011 Gabriele Pannocchia. |
| 5289 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
| 8 ## your option) any later version. | |
| 5289 | 9 ## |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 7016 | 16 ## along with Octave; see the file COPYING. If not, see |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 5289 | 18 |
| 19 ## -*- texinfo -*- | |
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20 ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}) |
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21 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}) |
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22 ## @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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23 ## @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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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}, @var{A_lb}, @var{A_in}, @var{A_ub}) |
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25 ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@dots{}, @var{options}) |
| 5289 | 26 ## Solve the quadratic program |
| 6741 | 27 ## @tex |
| 28 ## $$ | |
| 29 ## \min_x {1 \over 2} x^T H x + x^T q | |
| 30 ## $$ | |
| 31 ## @end tex | |
| 32 ## @ifnottex | |
| 5289 | 33 ## |
| 34 ## @example | |
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35 ## @group |
| 5289 | 36 ## min 0.5 x'*H*x + x'*q |
| 37 ## x | |
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38 ## @end group |
| 5289 | 39 ## @end example |
| 40 ## | |
| 6741 | 41 ## @end ifnottex |
| 42 ## subject to | |
| 5289 | 43 ## @tex |
| 6741 | 44 ## $$ |
| 45 ## Ax = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} \leq A_{ub} | |
| 46 ## $$ | |
| 5289 | 47 ## @end tex |
| 6741 | 48 ## @ifnottex |
| 5289 | 49 ## |
| 50 ## @example | |
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51 ## @group |
| 5289 | 52 ## A*x = b |
| 53 ## lb <= x <= ub | |
| 6741 | 54 ## A_lb <= A_in*x <= A_ub |
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55 ## @end group |
| 5289 | 56 ## @end example |
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57 ## |
| 6741 | 58 ## @end ifnottex |
| 5289 | 59 ## @noindent |
| 60 ## using a null-space active-set method. | |
| 61 ## | |
| 62 ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, | |
| 63 ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not | |
| 64 ## present. If the initial guess is feasible the algorithm is faster. | |
| 65 ## | |
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66 ## @table @var |
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67 ## @item options |
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68 ## An optional structure containing the following |
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69 ## parameter(s) used to define the behavior of the solver. Missing elements |
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70 ## in the structure take on default values, so you only need to set the |
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71 ## elements that you wish to change from the default. |
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72 ## |
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73 ## @table @code |
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74 ## @item MaxIter (default: 200) |
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75 ## Maximum number of iterations. |
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76 ## @end table |
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77 ## @end table |
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78 ## |
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79 ## @table @var |
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80 ## @item info |
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81 ## Structure containing run-time information about the algorithm. The |
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82 ## following fields are defined: |
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83 ## |
| 5289 | 84 ## @table @code |
| 85 ## @item solveiter | |
| 86 ## The number of iterations required to find the solution. | |
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87 ## |
| 5289 | 88 ## @item info |
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89 ## An integer indicating the status of the solution. |
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90 ## |
| 5289 | 91 ## @table @asis |
| 92 ## @item 0 | |
| 93 ## The problem is feasible and convex. Global solution found. | |
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94 ## |
| 5289 | 95 ## @item 1 |
| 96 ## The problem is not convex. Local solution found. | |
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97 ## |
| 5289 | 98 ## @item 2 |
| 99 ## The problem is not convex and unbounded. | |
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100 ## |
| 5289 | 101 ## @item 3 |
| 102 ## Maximum number of iterations reached. | |
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103 ## |
| 5289 | 104 ## @item 6 |
| 105 ## The problem is infeasible. | |
| 106 ## @end table | |
| 107 ## @end table | |
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108 ## @end table |
| 5289 | 109 ## @end deftypefn |
| 110 | |
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111 ## PKG_ADD: __all_opts__ ("qp"); |
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112 |
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113 function [x, obj, INFO, lambda] = qp (x0, H, varargin) |
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114 |
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115 nargs = nargin; |
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116 |
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117 if (nargin == 1 && ischar (x0) && strcmp (x0, 'defaults')) |
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118 x = optimset ("MaxIter", 200); |
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119 return; |
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120 endif |
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121 |
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122 if (nargs > 2 && isstruct (varargin{end})) |
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123 options = varargin{end}; |
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124 nargs--; |
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125 else |
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126 options = struct (); |
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127 endif |
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128 |
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129 if (nargs >= 3) |
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130 q = varargin{1}; |
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131 else |
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132 q = []; |
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133 endif |
| 5289 | 134 |
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135 if (nargs >= 5) |
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136 A = varargin{2}; |
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137 b = varargin{3}; |
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138 else |
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139 A = []; |
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140 b = []; |
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141 endif |
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142 |
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143 if (nargs >= 7) |
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144 lb = varargin{4}; |
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145 ub = varargin{5}; |
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146 else |
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147 lb = []; |
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148 ub = []; |
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149 endif |
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150 |
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151 if (nargs == 10) |
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152 A_lb = varargin{6}; |
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153 A_in = varargin{7}; |
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154 A_ub = varargin{8}; |
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155 else |
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156 A_lb = []; |
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157 A_in = []; |
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158 A_ub = []; |
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159 endif |
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160 |
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161 if (nargs == 2 || nargs == 3 || nargs == 5 || nargs == 7 || nargs == 10) |
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162 |
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163 maxit = optimget (options, "MaxIter", 200); |
| 5289 | 164 |
| 165 ## Checking the quadratic penalty | |
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166 if (! issquare (H)) |
| 5289 | 167 error ("qp: quadratic penalty matrix not square"); |
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168 elseif (! ishermitian (H)) |
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169 ## warning ("qp: quadratic penalty matrix not hermitian"); |
| 5289 | 170 H = (H + H')/2; |
| 171 endif | |
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172 n = rows (H); |
| 5289 | 173 |
| 174 ## Checking the initial guess (if empty it is resized to the | |
| 175 ## right dimension and filled with 0) | |
| 176 if (isempty (x0)) | |
| 177 x0 = zeros (n, 1); | |
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178 elseif (numel (x0) != n) |
| 5289 | 179 error ("qp: the initial guess has incorrect length"); |
| 180 endif | |
| 181 | |
| 182 ## Linear penalty. | |
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183 if (isempty (q)) |
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184 q = zeros (n, 1); |
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185 elseif (numel (q) != n) |
| 5289 | 186 error ("qp: the linear term has incorrect length"); |
| 187 endif | |
| 188 | |
| 189 ## Equality constraint matrices | |
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190 if (isempty (A) || isempty (b)) |
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191 A = zeros (0, n); |
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192 b = zeros (0, 1); |
| 5289 | 193 n_eq = 0; |
| 194 else | |
| 195 [n_eq, n1] = size (A); | |
| 196 if (n1 != n) | |
| 10549 | 197 error ("qp: equality constraint matrix has incorrect column dimension"); |
| 5289 | 198 endif |
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199 if (numel (b) != n_eq) |
| 10549 | 200 error ("qp: equality constraint matrix and vector have inconsistent dimension"); |
| 5289 | 201 endif |
| 202 endif | |
| 203 | |
| 204 ## Bound constraints | |
| 205 Ain = zeros (0, n); | |
| 206 bin = zeros (0, 1); | |
| 207 n_in = 0; | |
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208 if (nargs > 5) |
| 5289 | 209 if (! isempty (lb)) |
| 10549 | 210 if (numel (lb) != n) |
| 211 error ("qp: lower bound has incorrect length"); | |
| 212 elseif (isempty (ub)) | |
| 213 Ain = [Ain; eye(n)]; | |
| 214 bin = [bin; lb]; | |
| 215 endif | |
| 5289 | 216 endif |
| 217 | |
| 218 if (! isempty (ub)) | |
| 10549 | 219 if (numel (ub) != n) |
| 220 error ("qp: upper bound has incorrect length"); | |
| 221 elseif (isempty (lb)) | |
| 222 Ain = [Ain; -eye(n)]; | |
| 223 bin = [bin; -ub]; | |
| 224 endif | |
| 5289 | 225 endif |
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226 |
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227 if (! isempty (lb) && ! isempty (ub)) |
| 10549 | 228 rtol = sqrt (eps); |
| 229 for i = 1:n | |
| 230 if (abs(lb (i) - ub(i)) < rtol*(1 + max (abs (lb(i) + ub(i))))) | |
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231 ## These are actually an equality constraint |
| 10549 | 232 tmprow = zeros(1,n); |
| 233 tmprow(i) = 1; | |
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234 A = [A;tmprow]; |
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235 b = [b; 0.5*(lb(i) + ub(i))]; |
| 10549 | 236 n_eq = n_eq + 1; |
| 237 else | |
| 238 tmprow = zeros(1,n); | |
| 239 tmprow(i) = 1; | |
| 240 Ain = [Ain; tmprow; -tmprow]; | |
| 241 bin = [bin; lb(i); -ub(i)]; | |
| 242 n_in = n_in + 2; | |
| 243 endif | |
| 244 endfor | |
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245 endif |
| 5289 | 246 endif |
| 247 | |
| 248 ## Inequality constraints | |
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249 if (nargs > 7) |
| 5289 | 250 [dimA_in, n1] = size (A_in); |
| 251 if (n1 != n) | |
| 10549 | 252 error ("qp: inequality constraint matrix has incorrect column dimension"); |
| 5289 | 253 else |
| 10549 | 254 if (! isempty (A_lb)) |
| 255 if (numel (A_lb) != dimA_in) | |
| 256 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); | |
| 257 elseif (isempty (A_ub)) | |
| 258 Ain = [Ain; A_in]; | |
| 259 bin = [bin; A_lb]; | |
| 260 endif | |
| 261 endif | |
| 262 if (! isempty (A_ub)) | |
| 263 if (numel (A_ub) != dimA_in) | |
| 264 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); | |
| 265 elseif (isempty (A_lb)) | |
| 266 Ain = [Ain; -A_in]; | |
| 267 bin = [bin; -A_ub]; | |
| 268 endif | |
| 269 endif | |
| 270 | |
| 271 if (! isempty (A_lb) && ! isempty (A_ub)) | |
| 272 rtol = sqrt (eps); | |
| 273 for i = 1:dimA_in | |
| 274 if (abs (A_lb(i) - A_ub(i)) < rtol*(1 + max (abs (A_lb(i) + A_ub(i))))) | |
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275 ## These are actually an equality constraint |
| 10549 | 276 tmprow = A_in(i,:); |
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277 A = [A;tmprow]; |
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278 b = [b; 0.5*(A_lb(i) + A_ub(i))]; |
| 10549 | 279 n_eq = n_eq + 1; |
| 280 else | |
| 281 tmprow = A_in(i,:); | |
| 282 Ain = [Ain; tmprow; -tmprow]; | |
| 283 bin = [bin; A_lb(i); -A_ub(i)]; | |
| 284 n_in = n_in + 2; | |
| 285 endif | |
| 286 endfor | |
| 287 endif | |
| 5289 | 288 endif |
| 289 endif | |
| 290 | |
| 291 ## Now we should have the following QP: | |
| 292 ## | |
| 293 ## min_x 0.5*x'*H*x + x'*q | |
| 294 ## s.t. A*x = b | |
| 295 ## Ain*x >= bin | |
| 296 | |
| 297 ## Discard inequality constraints that have -Inf bounds since those | |
| 298 ## will never be active. | |
| 299 idx = isinf (bin) & bin < 0; | |
| 6523 | 300 |
| 6526 | 301 bin(idx) = []; |
| 302 Ain(idx,:) = []; | |
| 5289 | 303 |
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304 n_in = numel (bin); |
| 5310 | 305 |
| 5289 | 306 ## Check if the initial guess is feasible. |
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307 if (isa (x0, "single") || isa (H, "single") || isa (q, "single") || isa (A, "single") |
| 10549 | 308 || isa (b, "single")) |
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309 rtol = sqrt (eps ("single")); |
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310 else |
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311 rtol = sqrt (eps); |
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312 endif |
| 5289 | 313 |
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314 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+abs (b))); |
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315 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+abs (bin)))); |
| 5289 | 316 |
| 317 info = 0; | |
| 318 if (eq_infeasible || in_infeasible) | |
| 319 ## The initial guess is not feasible. | |
| 320 ## First define xbar that is feasible with respect to the equality | |
| 321 ## constraints. | |
| 322 if (eq_infeasible) | |
| 10549 | 323 if (rank (A) < n_eq) |
| 324 error ("qp: equality constraint matrix must be full row rank") | |
| 325 endif | |
| 326 xbar = pinv (A) * b; | |
| 5289 | 327 else |
| 10549 | 328 xbar = x0; |
| 5289 | 329 endif |
| 330 | |
| 331 ## Check if xbar is feasible with respect to the inequality | |
| 332 ## constraints also. | |
| 333 if (n_in > 0) | |
| 10549 | 334 res = Ain * xbar - bin; |
| 335 if (any (res < -rtol * (1 + abs (bin)))) | |
| 336 ## xbar is not feasible with respect to the inequality | |
| 337 ## constraints. Compute a step in the null space of the | |
| 338 ## equality constraints, by solving a QP. If the slack is | |
| 339 ## small, we have a feasible initial guess. Otherwise, the | |
| 340 ## problem is infeasible. | |
| 341 if (n_eq > 0) | |
| 342 Z = null (A); | |
| 343 if (isempty (Z)) | |
| 344 ## The problem is infeasible because A is square and full | |
| 345 ## rank, but xbar is not feasible. | |
| 346 info = 6; | |
| 347 endif | |
| 348 endif | |
| 5289 | 349 |
| 10549 | 350 if (info != 6) |
| 5289 | 351 ## Solve an LP with additional slack variables to find |
| 10549 | 352 ## a feasible starting point. |
| 353 gamma = eye (n_in); | |
| 354 if (n_eq > 0) | |
| 355 Atmp = [Ain*Z, gamma]; | |
| 356 btmp = -res; | |
| 357 else | |
| 358 Atmp = [Ain, gamma]; | |
| 359 btmp = bin; | |
| 360 endif | |
| 361 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; | |
| 362 lb = [-Inf(n-n_eq,1); zeros(n_in,1)]; | |
| 363 ub = []; | |
| 364 ctype = repmat ("L", n_in, 1); | |
| 365 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); | |
| 366 if ((status == 180 || status == 181 || status == 151) | |
| 367 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) | |
| 368 ## We found a feasible starting point | |
| 369 if (n_eq > 0) | |
| 370 x0 = xbar + Z*P(1:n-n_eq); | |
| 371 else | |
| 372 x0 = P(1:n); | |
| 5289 | 373 endif |
| 10549 | 374 else |
| 375 ## The problem is infeasible | |
| 376 info = 6; | |
| 377 endif | |
| 378 endif | |
| 379 else | |
| 380 ## xbar is feasible. We use it a starting point. | |
| 381 x0 = xbar; | |
| 382 endif | |
| 5289 | 383 else |
| 10549 | 384 ## xbar is feasible. We use it a starting point. |
| 385 x0 = xbar; | |
| 5289 | 386 endif |
| 387 endif | |
| 388 | |
| 389 if (info == 0) | |
| 390 ## The initial (or computed) guess is feasible. | |
| 391 ## We call the solver. | |
| 392 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); | |
| 393 else | |
| 394 iter = 0; | |
| 395 x = x0; | |
| 396 lambda = []; | |
| 397 endif | |
| 398 obj = 0.5 * x' * H * x + q' * x; | |
| 399 INFO.solveiter = iter; | |
| 400 INFO.info = info; | |
| 401 | |
| 402 else | |
| 6046 | 403 print_usage (); |
| 5289 | 404 endif |
| 405 | |
| 406 endfunction |