Mercurial > octave
annotate scripts/optimization/sqp.m @ 9051:1bf0ce0930be
Grammar check TexInfo in all .m files
Cleanup documentation sources to follow a few consistent rules.
Spellcheck was NOT done. (but will be in another changeset)
| author | Rik <rdrider0-list@yahoo.com> |
|---|---|
| date | Fri, 27 Mar 2009 22:31:03 -0700 |
| parents | eb63fbe60fab |
| children | 923c7cb7f13f |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2005, 2006, 2007, 2008 John W. Eaton |
| 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 -*- | |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
20 ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{iter}, @var{nf}, @var{lambda}] =} sqp (@var{x}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter}, @var{tolerance}) |
| 5289 | 21 ## Solve the nonlinear program |
| 6741 | 22 ## @iftex |
| 23 ## @tex | |
| 24 ## $$ | |
| 25 ## \min_x \phi (x) | |
| 26 ## $$ | |
| 27 ## @end tex | |
| 28 ## @end iftex | |
| 29 ## @ifnottex | |
| 5289 | 30 ## |
| 31 ## @example | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
32 ## @group |
| 5289 | 33 ## min phi (x) |
| 34 ## x | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
35 ## @end group |
| 5289 | 36 ## @end example |
| 37 ## | |
| 6741 | 38 ## @end ifnottex |
| 39 ## subject to | |
| 5289 | 40 ## @iftex |
| 41 ## @tex | |
| 6741 | 42 ## $$ |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
43 ## g(x) = 0 \qquad h(x) \geq 0 \qquad lb \leq x \leq ub |
| 6741 | 44 ## $$ |
| 5289 | 45 ## @end tex |
| 46 ## @end iftex | |
| 6741 | 47 ## @ifnottex |
| 5289 | 48 ## |
| 49 ## @example | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
50 ## @group |
| 5289 | 51 ## g(x) = 0 |
| 52 ## h(x) >= 0 | |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
53 ## lb <= x <= ub |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
54 ## @end group |
| 5289 | 55 ## @end example |
| 6741 | 56 ## @end ifnottex |
| 5289 | 57 ## |
| 58 ## @noindent | |
| 59 ## using a successive quadratic programming method. | |
| 60 ## | |
| 61 ## The first argument is the initial guess for the vector @var{x}. | |
| 62 ## | |
| 7001 | 63 ## The second argument is a function handle pointing to the objective |
| 5289 | 64 ## function. The objective function must be of the form |
| 65 ## | |
| 66 ## @example | |
| 67 ## y = phi (x) | |
| 68 ## @end example | |
| 69 ## | |
| 70 ## @noindent | |
| 71 ## in which @var{x} is a vector and @var{y} is a scalar. | |
| 72 ## | |
| 73 ## The second argument may also be a 2- or 3-element cell array of | |
| 74 ## function handles. The first element should point to the objective | |
| 75 ## function, the second should point to a function that computes the | |
| 76 ## gradient of the objective function, and the third should point to a | |
| 77 ## function to compute the hessian of the objective function. If the | |
| 78 ## gradient function is not supplied, the gradient is computed by finite | |
| 79 ## differences. If the hessian function is not supplied, a BFGS update | |
| 80 ## formula is used to approximate the hessian. | |
| 81 ## | |
| 82 ## If supplied, the gradient function must be of the form | |
| 83 ## | |
| 84 ## @example | |
| 7031 | 85 ## g = gradient (x) |
| 5289 | 86 ## @end example |
| 87 ## | |
| 88 ## @noindent | |
| 89 ## in which @var{x} is a vector and @var{g} is a vector. | |
| 90 ## | |
| 91 ## If supplied, the hessian function must be of the form | |
| 92 ## | |
| 93 ## @example | |
| 7031 | 94 ## h = hessian (x) |
| 5289 | 95 ## @end example |
| 96 ## | |
| 97 ## @noindent | |
| 98 ## in which @var{x} is a vector and @var{h} is a matrix. | |
| 99 ## | |
| 100 ## The third and fourth arguments are function handles pointing to | |
| 101 ## functions that compute the equality constraints and the inequality | |
| 102 ## constraints, respectively. | |
| 103 ## | |
| 104 ## If your problem does not have equality (or inequality) constraints, | |
| 105 ## you may pass an empty matrix for @var{cef} (or @var{cif}). | |
| 106 ## | |
| 107 ## If supplied, the equality and inequality constraint functions must be | |
| 108 ## of the form | |
| 109 ## | |
| 110 ## @example | |
| 7031 | 111 ## r = f (x) |
| 5289 | 112 ## @end example |
| 113 ## | |
| 114 ## @noindent | |
| 115 ## in which @var{x} is a vector and @var{r} is a vector. | |
| 116 ## | |
| 117 ## The third and fourth arguments may also be 2-element cell arrays of | |
| 118 ## function handles. The first element should point to the constraint | |
| 119 ## function and the second should point to a function that computes the | |
| 120 ## gradient of the constraint function: | |
| 121 ## | |
| 6741 | 122 ## @iftex |
| 123 ## @tex | |
| 124 ## $$ | |
| 125 ## \Bigg( {\partial f(x) \over \partial x_1}, | |
| 126 ## {\partial f(x) \over \partial x_2}, \ldots, | |
| 127 ## {\partial f(x) \over \partial x_N} \Bigg)^T | |
| 128 ## $$ | |
| 129 ## @end tex | |
| 130 ## @end iftex | |
| 131 ## @ifnottex | |
| 5289 | 132 ## @example |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
133 ## @group |
| 5289 | 134 ## [ d f(x) d f(x) d f(x) ] |
| 135 ## transpose ( [ ------ ----- ... ------ ] ) | |
| 136 ## [ dx_1 dx_2 dx_N ] | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
137 ## @end group |
| 5289 | 138 ## @end example |
| 6741 | 139 ## @end ifnottex |
| 5289 | 140 ## |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
141 ## The fifth and sixth arguments are vectors containing lower and upper bounds |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
142 ## on @var{x}. These must be consistent with equality and inequality |
|
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
143 ## constraints @var{g} and @var{h}. If the bounds are not specified, or are |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
144 ## empty, they are set to -@var{realmax} and @var{realmax} by default. |
|
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
145 ## |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
146 ## The seventh argument is max. number of iterations. If not specified, |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
147 ## the default value is 100. |
|
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
148 ## |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
149 ## The eighth argument is tolerance for stopping criteria. If not specified, |
|
8167
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
150 ## the default value is @var{eps}. |
|
17352ccd860e
describe additional arguments in sqp() documentation string
Ivan Sutoris <ivan.sutoris@gmail.com>
parents:
8047
diff
changeset
|
151 ## |
| 5289 | 152 ## Here is an example of calling @code{sqp}: |
| 153 ## | |
| 154 ## @example | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
155 ## @group |
| 7031 | 156 ## function r = g (x) |
| 157 ## r = [ sumsq(x)-10; | |
| 158 ## x(2)*x(3)-5*x(4)*x(5); | |
| 159 ## x(1)^3+x(2)^3+1 ]; | |
| 160 ## endfunction | |
| 161 ## | |
| 162 ## function obj = phi (x) | |
| 163 ## obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2; | |
| 164 ## endfunction | |
| 5289 | 165 ## |
| 7031 | 166 ## x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; |
| 167 ## | |
| 168 ## [x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, []) | |
| 5289 | 169 ## |
| 7031 | 170 ## x = |
| 171 ## | |
| 172 ## -1.71714 | |
| 173 ## 1.59571 | |
| 174 ## 1.82725 | |
| 175 ## -0.76364 | |
| 176 ## -0.76364 | |
| 5289 | 177 ## |
| 7031 | 178 ## obj = 0.053950 |
| 179 ## info = 101 | |
| 180 ## iter = 8 | |
| 181 ## nf = 10 | |
| 182 ## lambda = | |
| 183 ## | |
| 184 ## -0.0401627 | |
| 185 ## 0.0379578 | |
| 186 ## -0.0052227 | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
187 ## @end group |
| 5289 | 188 ## @end example |
| 189 ## | |
| 190 ## The value returned in @var{info} may be one of the following: | |
| 191 ## @table @asis | |
| 192 ## @item 101 | |
| 193 ## The algorithm terminated because the norm of the last step was less | |
| 194 ## than @code{tol * norm (x))} (the value of tol is currently fixed at | |
| 195 ## @code{sqrt (eps)}---edit @file{sqp.m} to modify this value. | |
| 196 ## @item 102 | |
| 197 ## The BFGS update failed. | |
| 198 ## @item 103 | |
| 199 ## The maximum number of iterations was reached (the maximum number of | |
| 200 ## allowed iterations is currently fixed at 100---edit @file{sqp.m} to | |
| 201 ## increase this value). | |
| 202 ## @end table | |
| 5642 | 203 ## @seealso{qp} |
| 5289 | 204 ## @end deftypefn |
| 205 | |
| 6768 | 206 function [x, obj, info, iter, nf, lambda] = sqp (x, objf, cef, cif, lb, ub, maxiter, tolerance) |
| 5289 | 207 |
| 6768 | 208 global __sqp_nfun__; |
| 5289 | 209 global __sqp_obj_fun__; |
| 210 global __sqp_ce_fun__; | |
| 211 global __sqp_ci_fun__; | |
| 6768 | 212 global __sqp_cif__; |
| 213 global __sqp_cifcn__; | |
| 5289 | 214 |
| 6768 | 215 if (nargin >= 2 && nargin <= 8 && nargin != 5) |
| 5289 | 216 |
| 217 ## Choose an initial NxN symmetric positive definite Hessan | |
| 218 ## approximation B. | |
| 219 | |
| 220 n = length (x); | |
| 221 | |
| 222 ## Evaluate objective function, constraints, and gradients at initial | |
| 223 ## value of x. | |
| 224 ## | |
| 225 ## obj_fun | |
| 226 ## obj_grad | |
| 227 ## ce_fun -- equality constraint functions | |
| 228 ## ci_fun -- inequality constraint functions | |
| 229 ## A == [grad_{x_1} cx_fun, grad_{x_2} cx_fun, ..., grad_{x_n} cx_fun]^T | |
| 230 | |
| 231 obj_grd = @fd_obj_grd; | |
| 232 have_hess = 0; | |
| 233 if (iscell (objf)) | |
| 234 if (length (objf) > 0) | |
| 235 __sqp_obj_fun__ = obj_fun = objf{1}; | |
| 236 if (length (objf) > 1) | |
| 237 obj_grd = objf{2}; | |
| 238 if (length (objf) > 2) | |
| 239 obj_hess = objf{3}; | |
| 240 have_hess = 1; | |
| 241 endif | |
| 242 endif | |
| 243 else | |
| 244 error ("sqp: invalid objective function"); | |
| 245 endif | |
| 246 else | |
| 247 __sqp_obj_fun__ = obj_fun = objf; | |
| 248 endif | |
| 249 | |
| 250 ce_fun = @empty_cf; | |
| 251 ce_grd = @empty_jac; | |
| 252 if (nargin > 2) | |
| 253 ce_grd = @fd_ce_jac; | |
| 254 if (iscell (cef)) | |
| 255 if (length (cef) > 0) | |
| 256 __sqp_ce_fun__ = ce_fun = cef{1}; | |
| 257 if (length (cef) > 1) | |
| 258 ce_grd = cef{2}; | |
| 259 endif | |
| 260 else | |
| 261 error ("sqp: invalid equality constraint function"); | |
| 262 endif | |
| 263 elseif (! isempty (cef)) | |
| 264 ce_fun = cef; | |
| 265 endif | |
| 266 endif | |
| 267 __sqp_ce_fun__ = ce_fun; | |
| 268 | |
| 269 ci_fun = @empty_cf; | |
| 270 ci_grd = @empty_jac; | |
| 6768 | 271 |
| 5289 | 272 if (nargin > 3) |
| 6768 | 273 ## constraint function given by user with possibly gradient |
| 274 __sqp_cif__ = cif; | |
| 275 ## constraint function given by user without gradient | |
| 276 __sqp_cifcn__ = @empty_cf; | |
| 277 if (iscell (__sqp_cif__)) | |
| 278 if (length (__sqp_cif__) > 0) | |
| 279 __sqp_cifcn__ = __sqp_cif__{1}; | |
| 280 endif | |
| 281 elseif (! isempty (__sqp_cif__)) | |
| 282 __sqp_cifcn__ = __sqp_cif__; | |
| 283 endif | |
| 284 | |
| 285 if (nargin < 5) | |
| 286 ci_grd = @fd_ci_jac; | |
| 287 if (iscell (cif)) | |
| 288 if (length (cif) > 0) | |
| 289 __sqp_ci_fun__ = ci_fun = cif{1}; | |
| 290 if (length (cif) > 1) | |
| 291 ci_grd = cif{2}; | |
| 292 endif | |
| 293 else | |
| 294 error ("sqp: invalid equality constraint function"); | |
| 5289 | 295 endif |
| 6768 | 296 elseif (! isempty (cif)) |
| 297 ci_fun = cif; | |
| 298 endif | |
| 299 else | |
| 300 global __sqp_lb__; | |
| 301 if (isvector (lb)) | |
| 302 __sqp_lb__ = lb; | |
| 303 elseif (isempty (lb)) | |
|
7795
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
304 if (isa (x, "single")) |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
305 __sqp_lb__ = -realmax ("single"); |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
306 else |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
307 __sqp_lb__ = -realmax; |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
308 endif |
| 5289 | 309 else |
| 6768 | 310 error ("sqp: invalid lower bound"); |
| 5289 | 311 endif |
| 6768 | 312 |
| 313 global __sqp_ub__; | |
| 314 if (isvector (ub)) | |
| 315 __sqp_ub__ = ub; | |
| 316 elseif (isempty (lb)) | |
|
7795
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
317 if (isa (x, "single")) |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
318 __sqp_ub__ = realmax ("single"); |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
319 else |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
320 __sqp_ub__ = realmax; |
|
df9519e9990c
Handle single precision eps values
David Bateman <dbateman@free.fr>
parents:
7404
diff
changeset
|
321 endif |
| 6768 | 322 else |
| 323 error ("sqp: invalid upper bound"); | |
| 324 endif | |
| 325 | |
| 326 if (lb > ub) | |
| 327 error ("sqp: upper bound smaller than lower bound"); | |
| 328 endif | |
| 329 __sqp_ci_fun__ = ci_fun = @cf_ub_lb; | |
| 330 ci_grd = @cigrad_ub_lb; | |
| 331 endif | |
| 332 __sqp_ci_fun__ = ci_fun; | |
| 333 endif | |
| 334 | |
| 335 iter_max = 100; | |
| 336 if (nargin > 6 && ! isempty (maxiter)) | |
| 6921 | 337 if (isscalar (maxiter) && maxiter > 0 && round (maxiter) == maxiter) |
| 6768 | 338 iter_max = maxiter; |
| 339 else | |
| 340 error ("sqp: invalid number of maximum iterations"); | |
| 5289 | 341 endif |
| 342 endif | |
| 343 | |
| 6768 | 344 tol = sqrt (eps); |
| 345 if (nargin > 7 && ! isempty (tolerance)) | |
| 346 if (isscalar (tolerance) && tolerance > 0) | |
| 347 tol = tolerance; | |
| 348 else | |
| 349 error ("sqp: invalid value for tolerance"); | |
| 350 endif | |
| 351 endif | |
| 5289 | 352 |
| 353 iter = 0; | |
| 354 | |
| 355 obj = feval (obj_fun, x); | |
| 6768 | 356 __sqp_nfun__ = 1; |
| 5289 | 357 |
| 358 c = feval (obj_grd, x); | |
| 359 | |
| 6382 | 360 if (have_hess) |
| 361 B = feval (obj_hess, x); | |
| 362 else | |
| 363 B = eye (n, n); | |
| 364 endif | |
| 365 | |
| 5289 | 366 ce = feval (ce_fun, x); |
| 367 F = feval (ce_grd, x); | |
| 368 | |
| 369 ci = feval (ci_fun, x); | |
| 370 C = feval (ci_grd, x); | |
| 371 | |
| 372 A = [F; C]; | |
| 373 | |
| 374 ## Choose an initial lambda (x is provided by the caller). | |
| 375 | |
| 376 lambda = 100 * ones (rows (A), 1); | |
| 377 | |
| 378 qp_iter = 1; | |
| 379 alpha = 1; | |
| 380 | |
| 381 ## report (); | |
| 382 | |
| 6768 | 383 ## report (iter, qp_iter, alpha, __sqp_nfun__, obj); |
| 5289 | 384 |
| 6527 | 385 info = 0; |
| 386 | |
| 5289 | 387 while (++iter < iter_max) |
| 388 | |
| 389 ## Check convergence. This is just a simple check on the first | |
| 390 ## order necessary conditions. | |
| 391 | |
| 392 ## IDX is the indices of the active inequality constraints. | |
| 393 | |
| 394 nr_f = rows (F); | |
| 395 | |
| 396 lambda_e = lambda((1:nr_f)'); | |
| 397 lambda_i = lambda((nr_f+1:end)'); | |
| 398 | |
| 399 con = [ce; ci]; | |
| 400 | |
| 401 t0 = norm (c - A' * lambda); | |
| 402 t1 = norm (ce); | |
| 403 t2 = all (ci >= 0); | |
| 404 t3 = all (lambda_i >= 0); | |
| 405 t4 = norm (lambda .* con); | |
| 406 | |
| 407 if (t2 && t3 && max ([t0; t1; t4]) < tol) | |
| 408 break; | |
| 409 endif | |
| 410 | |
| 411 ## Compute search direction p by solving QP. | |
| 412 | |
| 413 g = -ce; | |
| 414 d = -ci; | |
| 415 | |
| 416 ## Discard inequality constraints that have -Inf bounds since those | |
| 417 ## will never be active. | |
| 418 idx = isinf (d) & d < 0; | |
| 419 d(idx) = []; | |
| 420 C(idx,:) = []; | |
| 421 | |
| 422 [p, obj_qp, INFO, lambda] = qp (x, B, c, F, g, [], [], d, C, | |
| 423 Inf * ones (size (d))); | |
| 424 | |
| 425 info = INFO.info; | |
| 426 | |
| 427 ## Check QP solution and attempt to recover if it has failed. | |
| 428 | |
| 429 ## Choose mu such that p is a descent direction for the chosen | |
| 430 ## merit function phi. | |
| 431 | |
| 432 [x_new, alpha, obj_new] = linesearch_L1 (x, p, obj_fun, obj_grd, | |
| 433 ce_fun, ci_fun, lambda, obj); | |
| 434 | |
| 435 ## Evaluate objective function, constraints, and gradients at | |
| 436 ## x_new. | |
| 437 | |
| 438 c_new = feval (obj_grd, x_new); | |
| 439 | |
| 440 ce_new = feval (ce_fun, x_new); | |
| 441 F_new = feval (ce_grd, x_new); | |
| 442 | |
| 443 ci_new = feval (ci_fun, x_new); | |
| 444 C_new = feval (ci_grd, x_new); | |
| 445 | |
| 446 A_new = [F_new; C_new]; | |
| 447 | |
| 448 ## Set | |
| 449 ## | |
| 450 ## s = alpha * p | |
| 451 ## y = grad_x L (x_new, lambda) - grad_x L (x, lambda}) | |
| 452 | |
| 453 y = c_new - c; | |
| 454 | |
| 455 if (! isempty (A)) | |
| 456 t = ((A_new - A)'*lambda); | |
| 457 y -= t; | |
| 458 endif | |
| 459 | |
| 460 delx = x_new - x; | |
| 461 | |
| 462 if (norm (delx) < tol * norm (x)) | |
| 463 info = 101; | |
| 464 break; | |
| 465 endif | |
| 466 | |
| 467 if (have_hess) | |
| 468 | |
| 469 B = feval (obj_hess, x); | |
| 470 | |
| 471 else | |
| 472 | |
| 473 ## Update B using a quasi-Newton formula. | |
| 474 | |
| 475 delxt = delx'; | |
| 476 | |
| 477 ## Damped BFGS. Or maybe we would actually want to use the Hessian | |
| 478 ## of the Lagrangian, computed directly. | |
| 479 | |
| 480 d1 = delxt*B*delx; | |
| 481 | |
| 482 t1 = 0.2 * d1; | |
| 483 t2 = delxt*y; | |
| 484 | |
| 485 if (t2 < t1) | |
| 486 theta = 0.8*d1/(d1 - t2); | |
| 487 else | |
| 488 theta = 1; | |
| 489 endif | |
| 490 | |
| 491 r = theta*y + (1-theta)*B*delx; | |
| 492 | |
| 493 d2 = delxt*r; | |
| 494 | |
| 495 if (d1 == 0 || d2 == 0) | |
| 496 info = 102; | |
| 497 break; | |
| 498 endif | |
| 499 | |
| 500 B = B - B*delx*delxt*B/d1 + r*r'/d2; | |
| 501 | |
| 502 endif | |
| 503 | |
| 504 x = x_new; | |
| 505 | |
| 506 obj = obj_new; | |
| 507 | |
| 508 c = c_new; | |
| 509 | |
| 510 ce = ce_new; | |
| 511 F = F_new; | |
| 512 | |
| 513 ci = ci_new; | |
| 514 C = C_new; | |
| 515 | |
| 516 A = A_new; | |
| 517 | |
| 6768 | 518 ## report (iter, qp_iter, alpha, __sqp_nfun__, obj); |
| 5289 | 519 |
| 520 endwhile | |
| 521 | |
| 522 if (iter >= iter_max) | |
| 523 info = 103; | |
| 524 endif | |
| 525 | |
| 6768 | 526 nf = __sqp_nfun__; |
| 5289 | 527 |
| 528 else | |
| 529 | |
| 6046 | 530 print_usage (); |
| 5289 | 531 |
| 532 endif | |
| 533 | |
| 7399 | 534 endfunction |
| 5289 | 535 |
| 536 | |
| 537 function [merit, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu) | |
| 538 | |
| 6768 | 539 global __sqp_nfun__; |
| 5289 | 540 |
| 541 ce = feval (ce_fun, x); | |
| 542 ci = feval (ci_fun, x); | |
| 543 | |
| 544 idx = ci < 0; | |
| 545 | |
| 546 con = [ce; ci(idx)]; | |
| 547 | |
| 548 if (isempty (obj)) | |
| 549 obj = feval (obj_fun, x); | |
| 6768 | 550 __sqp_nfun__++; |
| 5289 | 551 endif |
| 552 | |
| 553 merit = obj; | |
| 554 t = norm (con, 1) / mu; | |
| 555 | |
| 556 if (! isempty (t)) | |
| 557 merit += t; | |
| 558 endif | |
| 559 | |
| 7399 | 560 endfunction |
| 5289 | 561 |
| 562 | |
| 563 function [x_new, alpha, obj] = linesearch_L1 (x, p, obj_fun, obj_grd, | |
| 564 ce_fun, ci_fun, lambda, obj) | |
| 565 | |
| 566 ## Choose parameters | |
| 567 ## | |
| 568 ## eta in the range (0, 0.5) | |
| 569 ## tau in the range (0, 1) | |
| 570 | |
| 571 eta = 0.25; | |
| 572 tau = 0.5; | |
| 573 | |
| 574 delta_bar = sqrt (eps); | |
| 575 | |
| 576 if (isempty (lambda)) | |
| 577 mu = 1 / delta_bar; | |
| 578 else | |
| 579 mu = 1 / (norm (lambda, Inf) + delta_bar); | |
| 580 endif | |
| 581 | |
| 582 alpha = 1; | |
| 583 | |
| 584 c = feval (obj_grd, x); | |
| 585 ce = feval (ce_fun, x); | |
| 586 | |
| 587 [phi_x_mu, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu); | |
| 588 | |
| 589 D_phi_x_mu = c' * p; | |
| 590 d = feval (ci_fun, x); | |
| 591 ## only those elements of d corresponding | |
| 592 ## to violated constraints should be included. | |
| 593 idx = d < 0; | |
| 594 t = - norm ([ce; d(idx)], 1) / mu; | |
| 595 if (! isempty (t)) | |
| 596 D_phi_x_mu += t; | |
| 597 endif | |
| 598 | |
| 599 while (1) | |
| 600 [p1, obj] = phi_L1 ([], obj_fun, ce_fun, ci_fun, x+alpha*p, mu); | |
| 601 p2 = phi_x_mu+eta*alpha*D_phi_x_mu; | |
| 602 if (p1 > p2) | |
| 603 ## Reset alpha = tau_alpha * alpha for some tau_alpha in the | |
| 604 ## range (0, tau). | |
| 605 tau_alpha = 0.9 * tau; ## ?? | |
| 606 alpha = tau_alpha * alpha; | |
| 607 else | |
| 608 break; | |
| 609 endif | |
| 610 endwhile | |
| 611 | |
| 612 ## Set x_new = x + alpha * p; | |
| 613 | |
| 614 x_new = x + alpha * p; | |
| 615 | |
| 7399 | 616 endfunction |
| 5289 | 617 |
| 618 | |
| 619 function report (iter, qp_iter, alpha, nfun, obj) | |
| 620 | |
| 621 if (nargin == 0) | |
| 622 printf (" Itn ItQP Step Nfun Objective\n"); | |
| 623 else | |
| 624 printf ("%5d %4d %8.1g %5d %13.6e\n", iter, qp_iter, alpha, nfun, obj); | |
| 625 endif | |
| 626 | |
| 7399 | 627 endfunction |
| 5289 | 628 |
| 629 | |
| 630 function grd = fdgrd (f, x) | |
| 631 | |
| 632 if (! isempty (f)) | |
| 633 y0 = feval (f, x); | |
| 634 nx = length (x); | |
| 635 grd = zeros (nx, 1); | |
| 636 deltax = sqrt (eps); | |
| 637 for i = 1:nx | |
| 638 t = x(i); | |
| 639 x(i) += deltax; | |
| 640 grd(i) = (feval (f, x) - y0) / deltax; | |
| 641 x(i) = t; | |
| 642 endfor | |
| 643 else | |
| 644 grd = zeros (0, 1); | |
| 645 endif | |
| 646 | |
| 7399 | 647 endfunction |
| 5289 | 648 |
| 649 | |
| 650 function jac = fdjac (f, x) | |
| 6768 | 651 nx = length (x); |
| 5289 | 652 if (! isempty (f)) |
| 653 y0 = feval (f, x); | |
| 654 nf = length (y0); | |
| 655 nx = length (x); | |
| 656 jac = zeros (nf, nx); | |
| 657 deltax = sqrt (eps); | |
| 658 for i = 1:nx | |
| 659 t = x(i); | |
| 660 x(i) += deltax; | |
| 661 jac(:,i) = (feval (f, x) - y0) / deltax; | |
| 662 x(i) = t; | |
| 663 endfor | |
| 664 else | |
| 665 jac = zeros (0, nx); | |
| 666 endif | |
| 667 | |
| 7399 | 668 endfunction |
| 5289 | 669 |
| 670 | |
| 671 function grd = fd_obj_grd (x) | |
| 672 | |
| 673 global __sqp_obj_fun__; | |
| 674 | |
| 675 grd = fdgrd (__sqp_obj_fun__, x); | |
| 676 | |
| 7399 | 677 endfunction |
| 5289 | 678 |
| 679 | |
| 680 function res = empty_cf (x) | |
| 681 | |
| 682 res = zeros (0, 1); | |
| 683 | |
| 7399 | 684 endfunction |
| 5289 | 685 |
| 686 | |
| 687 function res = empty_jac (x) | |
| 688 | |
| 689 res = zeros (0, length (x)); | |
| 690 | |
| 7399 | 691 endfunction |
| 5289 | 692 |
| 693 | |
| 694 function jac = fd_ce_jac (x) | |
| 695 | |
| 696 global __sqp_ce_fun__; | |
| 697 | |
| 698 jac = fdjac (__sqp_ce_fun__, x); | |
| 699 | |
| 7399 | 700 endfunction |
| 5289 | 701 |
| 702 | |
| 703 function jac = fd_ci_jac (x) | |
| 704 | |
| 6768 | 705 global __sqp_cifcn__; |
| 706 ## __sqp_cifcn__ = constraint function without gradients and lb or ub | |
| 707 jac = fdjac (__sqp_cifcn__, x); | |
| 708 | |
| 7399 | 709 endfunction |
| 6768 | 710 |
| 7017 | 711 |
| 6768 | 712 function res = cf_ub_lb (x) |
| 5289 | 713 |
| 6768 | 714 ## combine constraint function with ub and lb |
| 715 global __sqp_cifcn__ __sqp_lb__ __sqp_ub__ | |
| 716 | |
| 717 res = [x-__sqp_lb__; __sqp_ub__-x]; | |
| 718 | |
| 719 if (! isempty (__sqp_cifcn__)) | |
| 720 res = [feval(__sqp_cifcn__,x); x-__sqp_lb__; __sqp_ub__-x]; | |
| 721 endif | |
| 5289 | 722 |
| 7399 | 723 endfunction |
| 6768 | 724 |
| 7017 | 725 |
| 6768 | 726 function res = cigrad_ub_lb (x) |
| 727 | |
| 728 global __sqp_cif__ | |
| 729 | |
| 730 res = [eye(numel(x)); -eye(numel(x))]; | |
| 731 | |
| 732 cigradfcn = @fd_ci_jac; | |
| 733 | |
| 734 if (iscell (__sqp_cif__) && length (__sqp_cif__) > 1) | |
| 735 cigradfcn = __sqp_cif__{2}; | |
| 736 endif | |
| 737 | |
| 738 if (! isempty (cigradfcn)) | |
| 739 res = [feval(cigradfcn,x); eye(numel(x)); -eye(numel(x))]; | |
| 740 endif | |
| 741 | |
| 7399 | 742 endfunction |
| 7361 | 743 |
| 7371 | 744 %!function r = g (x) |
| 745 %! r = [sumsq(x)-10; | |
| 746 %! x(2)*x(3)-5*x(4)*x(5); | |
| 747 %! x(1)^3+x(2)^3+1 ]; | |
| 7361 | 748 %! |
| 7371 | 749 %!function obj = phi (x) |
| 750 %! obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2; | |
| 7361 | 751 %! |
| 752 %!test | |
| 753 %! x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; | |
| 754 %! | |
| 7381 | 755 %! [x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, []); |
| 7361 | 756 %! |
| 757 %! x_opt = [-1.717143501952599; | |
| 758 %! 1.595709610928535; | |
| 759 %! 1.827245880097156; | |
| 760 %! -0.763643103133572; | |
| 761 %! -0.763643068453300]; | |
| 762 %! | |
| 7371 | 763 %! obj_opt = 0.0539498477702739; |
| 7361 | 764 %! |
|
8047
d54f113aa983
Increase test script tolerances.
Thomas Treichl <Thomas.Treichl@gmx.net>
parents:
7795
diff
changeset
|
765 %! assert (all (abs (x-x_opt) < 5*sqrt (eps)) && abs (obj-obj_opt) < sqrt (eps)); |