Mercurial > hg > octave-nkf
annotate scripts/optimization/qp.m @ 7795:df9519e9990c
Handle single precision eps values
| author | David Bateman <dbateman@free.fr> |
|---|---|
| date | Mon, 12 May 2008 22:57:11 +0200 |
| parents | a1dbe9d80eee |
| children | 5ee11a81688e |
| rev | line source |
|---|---|
| 7017 | 1 ## Copyright (C) 2000, 2001, 2004, 2005, 2006, 2007 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 -*- | |
| 6741 | 20 ## @deftypefn {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}) |
| 5289 | 21 ## Solve the quadratic program |
| 6741 | 22 ## @iftex |
| 23 ## @tex | |
| 24 ## $$ | |
| 25 ## \min_x {1 \over 2} x^T H x + x^T q | |
| 26 ## $$ | |
| 27 ## @end tex | |
| 28 ## @end iftex | |
| 29 ## @ifnottex | |
| 5289 | 30 ## |
| 31 ## @example | |
| 32 ## min 0.5 x'*H*x + x'*q | |
| 33 ## x | |
| 34 ## @end example | |
| 35 ## | |
| 6741 | 36 ## @end ifnottex |
| 37 ## subject to | |
| 5289 | 38 ## @iftex |
| 39 ## @tex | |
| 6741 | 40 ## $$ |
| 41 ## Ax = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} \leq A_{ub} | |
| 42 ## $$ | |
| 5289 | 43 ## @end tex |
| 44 ## @end iftex | |
| 6741 | 45 ## @ifnottex |
| 5289 | 46 ## |
| 47 ## @example | |
| 48 ## A*x = b | |
| 49 ## lb <= x <= ub | |
| 6741 | 50 ## A_lb <= A_in*x <= A_ub |
| 5289 | 51 ## @end example |
| 6741 | 52 ## @end ifnottex |
| 5289 | 53 ## |
| 54 ## @noindent | |
| 55 ## using a null-space active-set method. | |
| 56 ## | |
| 57 ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, | |
| 58 ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not | |
| 59 ## present. If the initial guess is feasible the algorithm is faster. | |
| 60 ## | |
| 61 ## The value @var{info} is a structure with the following fields: | |
| 62 ## @table @code | |
| 63 ## @item solveiter | |
| 64 ## The number of iterations required to find the solution. | |
| 65 ## @item info | |
| 66 ## An integer indicating the status of the solution, as follows: | |
| 67 ## @table @asis | |
| 68 ## @item 0 | |
| 69 ## The problem is feasible and convex. Global solution found. | |
| 70 ## @item 1 | |
| 71 ## The problem is not convex. Local solution found. | |
| 72 ## @item 2 | |
| 73 ## The problem is not convex and unbounded. | |
| 74 ## @item 3 | |
| 75 ## Maximum number of iterations reached. | |
| 76 ## @item 6 | |
| 77 ## The problem is infeasible. | |
| 78 ## @end table | |
| 79 ## @end table | |
| 80 ## @end deftypefn | |
| 81 | |
| 82 function [x, obj, INFO, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub) | |
| 83 | |
| 84 if (nargin == 5 || nargin == 7 || nargin == 10) | |
| 85 | |
| 86 ## Checking the quadratic penalty | |
| 87 n = issquare (H); | |
| 88 if (n == 0) | |
| 89 error ("qp: quadratic penalty matrix not square"); | |
| 90 endif | |
| 91 | |
| 92 n1 = issymmetric (H); | |
| 93 if (n1 == 0) | |
| 94 ## warning ("qp: quadratic penalty matrix not symmetric"); | |
| 95 H = (H + H')/2; | |
| 96 endif | |
| 97 | |
| 98 ## Checking the initial guess (if empty it is resized to the | |
| 99 ## right dimension and filled with 0) | |
| 100 if (isempty (x0)) | |
| 101 x0 = zeros (n, 1); | |
| 102 elseif (length (x0) != n) | |
| 103 error ("qp: the initial guess has incorrect length"); | |
| 104 endif | |
| 105 | |
| 106 ## Linear penalty. | |
| 107 if (length (q) != n) | |
| 108 error ("qp: the linear term has incorrect length"); | |
| 109 endif | |
| 110 | |
| 111 ## Equality constraint matrices | |
| 112 if (isempty (A) || isempty(b)) | |
| 113 n_eq = 0; | |
| 114 A = zeros (n_eq, n); | |
| 115 b = zeros (n_eq, 1); | |
| 116 else | |
| 117 [n_eq, n1] = size (A); | |
| 118 if (n1 != n) | |
| 119 error ("qp: equality constraint matrix has incorrect column dimension"); | |
| 120 endif | |
| 121 if (length (b) != n_eq) | |
| 122 error ("qp: equality constraint matrix and vector have inconsistent dimension"); | |
| 123 endif | |
| 124 endif | |
| 125 | |
| 126 ## Bound constraints | |
| 127 Ain = zeros (0, n); | |
| 128 bin = zeros (0, 1); | |
| 129 n_in = 0; | |
| 130 if (nargin > 5) | |
| 131 if (! isempty (lb)) | |
| 132 if (length(lb) != n) | |
| 133 error ("qp: lower bound has incorrect length"); | |
| 134 else | |
| 135 Ain = [Ain; eye(n)]; | |
| 136 bin = [bin; lb]; | |
| 137 endif | |
| 138 endif | |
| 139 | |
| 140 if (! isempty (ub)) | |
| 141 if (length (ub) != n) | |
| 142 error ("qp: upper bound has incorrect length"); | |
| 143 else | |
| 144 Ain = [Ain; -eye(n)]; | |
| 145 bin = [bin; -ub]; | |
| 146 endif | |
| 147 endif | |
| 148 endif | |
| 149 | |
| 150 ## Inequality constraints | |
| 151 if (nargin > 7) | |
| 152 [dimA_in, n1] = size (A_in); | |
| 153 if (n1 != n) | |
| 154 error ("qp: inequality constraint matrix has incorrect column dimension"); | |
| 155 else | |
| 156 if (! isempty (A_lb)) | |
| 157 if (length (A_lb) != dimA_in) | |
| 158 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); | |
| 159 else | |
| 160 Ain = [Ain; A_in]; | |
| 161 bin = [bin; A_lb]; | |
| 162 endif | |
| 163 endif | |
| 164 if (! isempty (A_ub)) | |
| 165 if (length (A_ub) != dimA_in) | |
| 166 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); | |
| 167 else | |
| 168 Ain = [Ain; -A_in]; | |
| 169 bin = [bin; -A_ub]; | |
| 170 endif | |
| 171 endif | |
| 172 endif | |
| 173 endif | |
| 174 | |
| 175 ## Now we should have the following QP: | |
| 176 ## | |
| 177 ## min_x 0.5*x'*H*x + x'*q | |
| 178 ## s.t. A*x = b | |
| 179 ## Ain*x >= bin | |
| 180 | |
| 181 ## Discard inequality constraints that have -Inf bounds since those | |
| 182 ## will never be active. | |
| 183 idx = isinf (bin) & bin < 0; | |
| 6523 | 184 |
| 6526 | 185 bin(idx) = []; |
| 186 Ain(idx,:) = []; | |
| 5289 | 187 |
| 5310 | 188 n_in = length (bin); |
| 189 | |
| 5289 | 190 ## Check if the initial guess is feasible. |
|
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191 if (isa (x0, "single") || isa (H, "single") || isa (q, "single") || isa (A, "single") |
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192 || isa (b, "single")) |
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193 rtol = sqrt (eps ("single")); |
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194 else |
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195 rtol = sqrt (eps); |
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196 endif |
| 5289 | 197 |
| 198 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+norm (b))); | |
| 199 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+norm (bin)))); | |
| 200 | |
| 201 info = 0; | |
| 202 if (eq_infeasible || in_infeasible) | |
| 203 ## The initial guess is not feasible. | |
| 204 ## First define xbar that is feasible with respect to the equality | |
| 205 ## constraints. | |
| 206 if (eq_infeasible) | |
| 207 if (rank (A) < n_eq) | |
| 208 error ("qp: equality constraint matrix must be full row rank") | |
| 209 endif | |
| 210 xbar = pinv (A) * b; | |
| 211 else | |
| 212 xbar = x0; | |
| 213 endif | |
| 214 | |
| 215 ## Check if xbar is feasible with respect to the inequality | |
| 216 ## constraints also. | |
| 217 if (n_in > 0) | |
| 218 res = Ain * xbar - bin; | |
| 219 if (any (res < -rtol * (1 + norm (bin)))) | |
| 220 ## xbar is not feasible with respect to the inequality | |
| 221 ## constraints. Compute a step in the null space of the | |
| 222 ## equality constraints, by solving a QP. If the slack is | |
| 223 ## small, we have a feasible initial guess. Otherwise, the | |
| 224 ## problem is infeasible. | |
| 225 if (n_eq > 0) | |
| 226 Z = null (A); | |
| 227 if (isempty (Z)) | |
| 228 ## The problem is infeasible because A is square and full | |
| 229 ## rank, but xbar is not feasible. | |
| 230 info = 6; | |
| 231 endif | |
| 232 endif | |
| 233 | |
| 234 if (info != 6) | |
| 235 ## Solve an LP with additional slack variables to find | |
| 236 ## a feasible starting point. | |
| 237 gamma = eye (n_in); | |
| 238 if (n_eq > 0) | |
| 239 Atmp = [Ain*Z, gamma]; | |
| 240 btmp = -res; | |
| 241 else | |
| 242 Atmp = [Ain, gamma]; | |
| 243 btmp = bin; | |
| 244 endif | |
| 245 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; | |
| 246 lb = [-Inf*ones(n-n_eq,1); zeros(n_in,1)]; | |
| 247 ub = []; | |
| 248 ctype = repmat ("L", n_in, 1); | |
| 249 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); | |
| 250 if ((status == 180 || status == 181 || status == 151) | |
| 251 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) | |
| 252 ## We found a feasible starting point | |
| 253 if (n_eq > 0) | |
| 5306 | 254 x0 = xbar + Z*P(1:n-n_eq); |
| 5289 | 255 else |
| 256 x0 = P(1:n); | |
| 257 endif | |
| 258 else | |
| 259 ## The problem is infeasible | |
| 260 info = 6; | |
| 261 endif | |
| 262 endif | |
| 263 else | |
| 264 ## xbar is feasible. We use it a starting point. | |
| 265 x0 = xbar; | |
| 266 endif | |
| 267 else | |
| 268 ## xbar is feasible. We use it a starting point. | |
| 269 x0 = xbar; | |
| 270 endif | |
| 271 endif | |
| 272 | |
| 273 if (info == 0) | |
| 6848 | 274 ## FIXME -- make maxit a user option. |
| 5289 | 275 ## The initial (or computed) guess is feasible. |
| 276 ## We call the solver. | |
| 6848 | 277 maxit = 200; |
| 5289 | 278 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); |
| 279 else | |
| 280 iter = 0; | |
| 281 x = x0; | |
| 282 lambda = []; | |
| 283 endif | |
| 284 obj = 0.5 * x' * H * x + q' * x; | |
| 285 INFO.solveiter = iter; | |
| 286 INFO.info = info; | |
| 287 | |
| 288 else | |
| 6046 | 289 print_usage (); |
| 5289 | 290 endif |
| 291 | |
| 292 endfunction |