Mercurial > hg > octave-nkf
annotate scripts/optimization/qp.m @ 8507:cadc73247d65
style fixes
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Tue, 13 Jan 2009 14:08:36 -0500 |
| parents | 5ee11a81688e |
| children | eb63fbe60fab |
| 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"); | |
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134 elseif (isempty (ub)) |
| 5289 | 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"); | |
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143 elseif (isempty (lb)) |
| 5289 | 144 Ain = [Ain; -eye(n)]; |
| 145 bin = [bin; -ub]; | |
| 146 endif | |
| 147 endif | |
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148 |
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149 if (! isempty (lb) && ! isempty (ub)) |
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150 rtol = sqrt (eps); |
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151 for i = 1:n |
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152 if (abs(lb (i) - ub(i)) < rtol*(1 + max (abs (lb(i) + ub(i))))) |
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153 ## These are actually an equality constraint |
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154 tmprow = zeros(1,n); |
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155 tmprow(i) = 1; |
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156 A = [A;tmprow]; |
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157 b = [b; 0.5*(lb(i) + ub(i))]; |
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158 n_eq = n_eq + 1; |
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159 else |
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160 tmprow = zeros(1,n); |
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161 tmprow(i) = 1; |
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162 Ain = [Ain; tmprow; -tmprow]; |
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163 bin = [bin; lb(i); -ub(i)]; |
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164 n_in = n_in + 2; |
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165 endif |
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166 endfor |
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167 endif |
| 5289 | 168 endif |
| 169 | |
| 170 ## Inequality constraints | |
| 171 if (nargin > 7) | |
| 172 [dimA_in, n1] = size (A_in); | |
| 173 if (n1 != n) | |
| 174 error ("qp: inequality constraint matrix has incorrect column dimension"); | |
| 175 else | |
| 176 if (! isempty (A_lb)) | |
| 177 if (length (A_lb) != dimA_in) | |
| 178 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); | |
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179 elseif (isempty (A_ub)) |
| 5289 | 180 Ain = [Ain; A_in]; |
| 181 bin = [bin; A_lb]; | |
| 182 endif | |
| 183 endif | |
| 184 if (! isempty (A_ub)) | |
| 185 if (length (A_ub) != dimA_in) | |
| 186 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); | |
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187 elseif (isempty (A_lb)) |
| 5289 | 188 Ain = [Ain; -A_in]; |
| 189 bin = [bin; -A_ub]; | |
| 190 endif | |
| 191 endif | |
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192 |
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193 if (! isempty (A_lb) && ! isempty (A_ub)) |
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194 rtol = sqrt (eps); |
| 8507 | 195 for i = 1:dimA_in |
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196 if (abs (A_lb(i) - A_ub(i)) < rtol*(1 + max (abs (A_lb(i) + A_ub(i))))) |
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197 ## These are actually an equality constraint |
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198 tmprow = A_in(i,:); |
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199 A = [A;tmprow]; |
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200 b = [b; 0.5*(A_lb(i) + A_ub(i))]; |
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201 n_eq = n_eq + 1; |
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202 else |
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203 tmprow = A_in(i,:); |
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204 Ain = [Ain; tmprow; -tmprow]; |
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205 bin = [bin; A_lb(i); -A_ub(i)]; |
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206 n_in = n_in + 2; |
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207 endif |
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208 endfor |
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209 endif |
| 5289 | 210 endif |
| 211 endif | |
| 212 | |
| 213 ## Now we should have the following QP: | |
| 214 ## | |
| 215 ## min_x 0.5*x'*H*x + x'*q | |
| 216 ## s.t. A*x = b | |
| 217 ## Ain*x >= bin | |
| 218 | |
| 219 ## Discard inequality constraints that have -Inf bounds since those | |
| 220 ## will never be active. | |
| 221 idx = isinf (bin) & bin < 0; | |
| 6523 | 222 |
| 6526 | 223 bin(idx) = []; |
| 224 Ain(idx,:) = []; | |
| 5289 | 225 |
| 5310 | 226 n_in = length (bin); |
| 227 | |
| 5289 | 228 ## Check if the initial guess is feasible. |
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229 if (isa (x0, "single") || isa (H, "single") || isa (q, "single") || isa (A, "single") |
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230 || isa (b, "single")) |
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231 rtol = sqrt (eps ("single")); |
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232 else |
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233 rtol = sqrt (eps); |
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234 endif |
| 5289 | 235 |
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236 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+abs (b))); |
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237 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+abs (bin)))); |
| 5289 | 238 |
| 239 info = 0; | |
| 240 if (eq_infeasible || in_infeasible) | |
| 241 ## The initial guess is not feasible. | |
| 242 ## First define xbar that is feasible with respect to the equality | |
| 243 ## constraints. | |
| 244 if (eq_infeasible) | |
| 245 if (rank (A) < n_eq) | |
| 246 error ("qp: equality constraint matrix must be full row rank") | |
| 247 endif | |
| 248 xbar = pinv (A) * b; | |
| 249 else | |
| 250 xbar = x0; | |
| 251 endif | |
| 252 | |
| 253 ## Check if xbar is feasible with respect to the inequality | |
| 254 ## constraints also. | |
| 255 if (n_in > 0) | |
| 256 res = Ain * xbar - bin; | |
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257 if (any (res < -rtol * (1 + abs (bin)))) |
| 5289 | 258 ## xbar is not feasible with respect to the inequality |
| 259 ## constraints. Compute a step in the null space of the | |
| 260 ## equality constraints, by solving a QP. If the slack is | |
| 261 ## small, we have a feasible initial guess. Otherwise, the | |
| 262 ## problem is infeasible. | |
| 263 if (n_eq > 0) | |
| 264 Z = null (A); | |
| 265 if (isempty (Z)) | |
| 266 ## The problem is infeasible because A is square and full | |
| 267 ## rank, but xbar is not feasible. | |
| 268 info = 6; | |
| 269 endif | |
| 270 endif | |
| 271 | |
| 272 if (info != 6) | |
| 273 ## Solve an LP with additional slack variables to find | |
| 274 ## a feasible starting point. | |
| 275 gamma = eye (n_in); | |
| 276 if (n_eq > 0) | |
| 277 Atmp = [Ain*Z, gamma]; | |
| 278 btmp = -res; | |
| 279 else | |
| 280 Atmp = [Ain, gamma]; | |
| 281 btmp = bin; | |
| 282 endif | |
| 283 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; | |
| 284 lb = [-Inf*ones(n-n_eq,1); zeros(n_in,1)]; | |
| 285 ub = []; | |
| 286 ctype = repmat ("L", n_in, 1); | |
| 287 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); | |
| 288 if ((status == 180 || status == 181 || status == 151) | |
| 289 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) | |
| 290 ## We found a feasible starting point | |
| 291 if (n_eq > 0) | |
| 5306 | 292 x0 = xbar + Z*P(1:n-n_eq); |
| 5289 | 293 else |
| 294 x0 = P(1:n); | |
| 295 endif | |
| 296 else | |
| 297 ## The problem is infeasible | |
| 298 info = 6; | |
| 299 endif | |
| 300 endif | |
| 301 else | |
| 302 ## xbar is feasible. We use it a starting point. | |
| 303 x0 = xbar; | |
| 304 endif | |
| 305 else | |
| 306 ## xbar is feasible. We use it a starting point. | |
| 307 x0 = xbar; | |
| 308 endif | |
| 309 endif | |
| 310 | |
| 311 if (info == 0) | |
| 6848 | 312 ## FIXME -- make maxit a user option. |
| 5289 | 313 ## The initial (or computed) guess is feasible. |
| 314 ## We call the solver. | |
| 6848 | 315 maxit = 200; |
| 5289 | 316 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); |
| 317 else | |
| 318 iter = 0; | |
| 319 x = x0; | |
| 320 lambda = []; | |
| 321 endif | |
| 322 obj = 0.5 * x' * H * x + q' * x; | |
| 323 INFO.solveiter = iter; | |
| 324 INFO.info = info; | |
| 325 | |
| 326 else | |
| 6046 | 327 print_usage (); |
| 5289 | 328 endif |
| 329 | |
| 330 endfunction |