Mercurial > octave
annotate scripts/optimization/fminsearch.m @ 23084:ef4d915df748
maint: Merge stable to default.
| author | John W. Eaton <jwe@octave.org> |
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| date | Mon, 23 Jan 2017 14:27:48 -0500 |
| parents | 3c896180df16 e9a0469dedd9 |
| children | dffa2b8c9482 |
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1 ## Copyright (C) 2003-2016 Andy Adler |
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2 ## Copyright (C) 2002, 2013 N.J.Higham |
| 14895 | 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 | |
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8 ## the Free Software Foundation; either version 3 of the License, or |
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9 ## (at your option) any later version. |
| 14895 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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14 ## GNU General Public License for more details. |
| 14895 | 15 ## |
| 16 ## You should have received a copy of the GNU General Public License | |
| 17 ## along with Octave; see the file COPYING. If not, see | |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 19 | |
| 20 ## -*- texinfo -*- | |
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21 ## @deftypefn {} {@var{x} =} fminsearch (@var{fun}, @var{x0}) |
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22 ## @deftypefnx {} {@var{x} =} fminsearch (@var{fun}, @var{x0}, @var{options}) |
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23 ## @deftypefnx {} {@var{x} =} fminsearch (@var{fun}, @var{x0}, @var{options}, @var{fun_arg1}, @var{fun_arg2}, @dots{}) |
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24 ## @deftypefnx {} {[@var{x}, @var{fval}] =} fminsearch (@dots{}) |
| 14895 | 25 ## |
| 26 ## Find a value of @var{x} which minimizes the function @var{fun}. | |
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27 ## |
| 14895 | 28 ## The search begins at the point @var{x0} and iterates using the |
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29 ## @nospell{Nelder & Mead} Simplex algorithm (a derivative-free method). This |
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30 ## algorithm is better-suited to functions which have discontinuities or for |
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31 ## which a gradient-based search such as @code{fminunc} fails. |
| 14895 | 32 ## |
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33 ## Options for the search are provided in the parameter @var{options} using the |
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34 ## function @code{optimset}. Currently, @code{fminsearch} accepts the options: |
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35 ## @qcode{"TolX"}, @qcode{"MaxFunEvals"}, @qcode{"MaxIter"}, @qcode{"Display"}. |
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36 ## For a description of these options, see @code{optimset}. |
| 14895 | 37 ## |
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38 ## Additional inputs for the function @var{fun} can be passed as the fourth |
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39 ## and higher arguments. To pass function arguments while using the default |
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40 ## @var{options} values, use @code{[]} for @var{options}. |
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41 ## |
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42 ## On exit, the function returns @var{x}, the minimum point, and @var{fval}, |
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43 ## the function value at the minimum. |
| 14895 | 44 ## |
| 45 ## Example usages: | |
| 46 ## | |
| 47 ## @example | |
| 48 ## @group | |
| 49 ## fminsearch (@@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0]) | |
| 50 ## | |
| 51 ## fminsearch (inline ("(x(1)-5).^2+(x(2)-8).^4", "x"), [0;0]) | |
| 52 ## @end group | |
| 53 ## @end example | |
| 54 ## @seealso{fminbnd, fminunc, optimset} | |
| 55 ## @end deftypefn | |
| 56 | |
| 57 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. | |
| 58 ## PKG_ADD: [~] = __all_opts__ ("fminsearch"); | |
| 59 | |
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60 ## FIXME: Add support for "exitflag" output variable. |
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61 ## FIXME: Add support for "output" output variable. |
| 14895 | 62 ## FIXME: For Display option, add 'final' and 'notify' options. Not too hard. |
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63 ## FIXME: Add support for OutputFcn. See fminunc for a template. |
| 14895 | 64 ## FIXME: Add support for exiting based on TolFun. See fminunc for an idea. |
| 65 | |
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66 function [x, fval] = fminsearch (fun, x0, options, varargin) |
| 14895 | 67 |
| 68 ## Get default options if requested. | |
| 69 if (nargin == 1 && ischar (fun) && strcmp (fun, "defaults")) | |
| 70 x = optimset ("Display", "notify", "FunValCheck", "off", | |
| 71 "MaxFunEvals", 400, "MaxIter", 400, | |
| 72 "OutputFcn", [], | |
| 73 "TolFun", 1e-7, "TolX", 1e-4); | |
| 74 return; | |
| 75 endif | |
| 76 | |
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77 if (nargin < 2) |
| 14895 | 78 print_usage (); |
| 79 endif | |
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80 |
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81 if (nargin < 3 || isempty (options)) |
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82 options = struct (); |
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83 endif |
| 14895 | 84 |
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85 x = nmsmax (fun, x0, options, varargin{:}); |
| 14895 | 86 |
| 87 if (isargout (2)) | |
| 88 fval = feval (fun, x); | |
| 89 endif | |
| 90 | |
| 91 endfunction | |
| 92 | |
| 93 ##NMSMAX Nelder-Mead simplex method for direct search optimization. | |
| 94 ## [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to | |
| 95 ## maximize the function FUN, using the starting vector x0. | |
| 96 ## The Nelder-Mead direct search method is used. | |
| 97 ## Output arguments: | |
| 98 ## x = vector yielding largest function value found, | |
| 99 ## fmax = function value at x, | |
| 100 ## nf = number of function evaluations. | |
| 101 ## The iteration is terminated when either | |
| 102 ## - the relative size of the simplex is <= STOPIT(1) | |
| 103 ## (default 1e-3), | |
| 104 ## - STOPIT(2) function evaluations have been performed | |
| 105 ## (default inf, i.e., no limit), or | |
| 106 ## - a function value equals or exceeds STOPIT(3) | |
| 107 ## (default inf, i.e., no test on function values). | |
| 108 ## The form of the initial simplex is determined by STOPIT(4): | |
| 109 ## STOPIT(4) = 0: regular simplex (sides of equal length, the default) | |
| 110 ## STOPIT(4) = 1: right-angled simplex. | |
| 111 ## Progress of the iteration is not shown if STOPIT(5) = 0 (default 1). | |
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112 ## STOPIT(6) indicates the direction (i.e., minimization or |
| 14895 | 113 ## maximization.) Default is 1, maximization. |
| 114 ## set STOPIT(6)=-1 for minimization | |
| 115 ## If a non-empty fourth parameter string SAVIT is present, then | |
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116 ## 'SAVE SAVIT x fmax nf' is executed after each inner iteration. |
| 14895 | 117 ## NB: x0 can be a matrix. In the output argument, in SAVIT saves, |
| 118 ## and in function calls, x has the same shape as x0. | |
| 119 ## NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional | |
| 120 ## arguments to be passed to fun, via feval(fun,x,P1,P2,...). | |
| 121 ## References: | |
| 122 ## N. J. Higham, Optimization by direct search in matrix computations, | |
| 123 ## SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. | |
| 124 ## C. T. Kelley, Iterative Methods for Optimization, Society for Industrial | |
| 125 ## and Applied Mathematics, Philadelphia, PA, 1999. | |
| 126 | |
| 127 ## From Matrix Toolbox | |
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128 ## Copyright (C) 2002, 2013 N.J.Higham |
| 14895 | 129 ## www.maths.man.ac.uk/~higham/mctoolbox |
| 130 ## | |
| 131 ## Modifications for Octave by A.Adler 2003 | |
| 132 | |
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133 function [stopit, savit, dirn, trace, tol, maxiter] = parse_options (options, x) |
| 14895 | 134 |
| 135 ## Tolerance for cgce test based on relative size of simplex. | |
| 136 stopit(1) = tol = optimget (options, "TolX", 1e-4); | |
| 137 | |
| 138 ## Max no. of f-evaluations. | |
| 139 stopit(2) = optimget (options, "MaxFunEvals", length (x) * 200); | |
| 140 | |
| 141 ## Max no. of iterations | |
| 142 maxiter = optimget (options, "MaxIter", length (x) * 200); | |
| 143 | |
| 144 ## Default target for f-values. | |
| 145 stopit(3) = Inf; # FIXME: expose this parameter to the outside | |
| 146 | |
| 147 ## Default initial simplex. | |
| 148 stopit(4) = 0; # FIXME: expose this parameter to the outside | |
| 149 | |
| 150 ## Default: show progress. | |
| 151 display = optimget (options, "Display", "notify"); | |
| 152 if (strcmp (display, "iter")) | |
| 153 stopit(5) = 1; | |
| 154 else | |
| 155 stopit(5) = 0; | |
| 156 endif | |
| 157 trace = stopit(5); | |
| 158 | |
| 159 ## Use function to minimize, not maximize | |
| 160 stopit(6) = dirn = -1; | |
| 161 | |
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162 ## Filename for snapshots. |
| 14895 | 163 savit = []; # FIXME: expose this parameter to the outside |
| 164 | |
| 165 endfunction | |
| 166 | |
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167 function [x, fmax, nf] = nmsmax (fun, x, options, varargin) |
| 14895 | 168 |
| 169 [stopit, savit, dirn, trace, tol, maxiter] = parse_options (options, x); | |
| 170 | |
| 171 if (strcmpi (optimget (options, "FunValCheck", "off"), "on")) | |
| 172 ## Replace fcn with a guarded version. | |
| 173 fun = @(x) guarded_eval (fun, x); | |
| 174 endif | |
| 175 | |
| 176 x0 = x(:); # Work with column vector internally. | |
| 177 n = length (x0); | |
| 178 | |
| 179 V = [zeros(n,1) eye(n)]; | |
| 180 f = zeros (n+1,1); | |
| 181 V(:,1) = x0; | |
| 182 f(1) = dirn * feval (fun,x,varargin{:}); | |
| 183 fmax_old = f(1); | |
| 184 | |
| 185 if (trace) | |
| 186 fprintf ("f(x0) = %9.4e\n", f(1)); | |
| 187 endif | |
| 188 | |
| 189 k = 0; m = 0; | |
| 190 | |
| 191 ## Set up initial simplex. | |
| 192 scale = max (norm (x0,Inf), 1); | |
| 193 if (stopit(4) == 0) | |
| 194 ## Regular simplex - all edges have same length. | |
| 195 ## Generated from construction given in reference [18, pp. 80-81] of [1]. | |
| 196 alpha = scale / (n*sqrt (2)) * [sqrt(n+1)-1+n, sqrt(n+1)-1]; | |
| 197 V(:,2:n+1) = (x0 + alpha(2)*ones (n,1)) * ones (1,n); | |
| 198 for j = 2:n+1 | |
| 199 V(j-1,j) = x0(j-1) + alpha(1); | |
| 200 x(:) = V(:,j); | |
| 201 f(j) = dirn * feval (fun,x,varargin{:}); | |
| 202 endfor | |
| 203 else | |
| 204 ## Right-angled simplex based on co-ordinate axes. | |
| 205 alpha = scale * ones(n+1,1); | |
| 206 for j=2:n+1 | |
| 207 V(:,j) = x0 + alpha(j)*V(:,j); | |
| 208 x(:) = V(:,j); | |
| 209 f(j) = dirn * feval (fun,x,varargin{:}); | |
| 210 endfor | |
| 211 endif | |
| 212 nf = n+1; | |
| 213 how = "initial "; | |
| 214 | |
| 215 [~,j] = sort (f); | |
| 216 j = j(n+1:-1:1); | |
| 217 f = f(j); | |
| 218 V = V(:,j); | |
| 219 | |
| 220 alpha = 1; beta = 1/2; gamma = 2; | |
| 221 | |
| 222 while (1) # Outer (and only) loop. | |
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223 k += 1; |
| 14895 | 224 |
| 225 if (k > maxiter) | |
| 226 msg = "Exceeded maximum iterations...quitting\n"; | |
| 227 break; | |
| 228 endif | |
| 229 | |
| 230 fmax = f(1); | |
| 231 if (fmax > fmax_old) | |
| 232 if (! isempty (savit)) | |
| 233 x(:) = V(:,1); | |
| 234 eval (["save " savit " x fmax nf"]); | |
| 235 endif | |
| 236 endif | |
| 237 if (trace) | |
| 238 fprintf ("Iter. %2.0f,", k); | |
| 239 fprintf ([" how = " how " "]); | |
| 240 fprintf ("nf = %3.0f, f = %9.4e (%2.1f%%)\n", nf, fmax, ... | |
| 241 100*(fmax-fmax_old)/(abs(fmax_old)+eps)); | |
| 242 endif | |
| 243 fmax_old = fmax; | |
| 244 | |
| 245 ## Three stopping tests from MDSMAX.M | |
| 246 | |
| 247 ## Stopping Test 1 - f reached target value? | |
| 248 if (fmax >= stopit(3)) | |
| 249 msg = "Exceeded target...quitting\n"; | |
| 250 break; | |
| 251 endif | |
| 252 | |
| 253 ## Stopping Test 2 - too many f-evals? | |
| 254 if (nf >= stopit(2)) | |
| 255 msg = "Max no. of function evaluations exceeded...quitting\n"; | |
| 256 break; | |
| 257 endif | |
| 258 | |
| 259 ## Stopping Test 3 - converged? This is test (4.3) in [1]. | |
| 260 v1 = V(:,1); | |
| 261 size_simplex = norm (V(:,2:n+1)-v1(:,ones (1,n)),1) / max (1, norm (v1,1)); | |
| 262 if (size_simplex <= tol) | |
| 263 msg = sprintf ("Simplex size %9.4e <= %9.4e...quitting\n", ... | |
| 264 size_simplex, tol); | |
| 265 break; | |
| 266 endif | |
| 267 | |
| 268 ## One step of the Nelder-Mead simplex algorithm | |
| 269 ## NJH: Altered function calls and changed CNT to NF. | |
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270 ## Changed each 'fr < f(1)' type test to '>' for maximization |
| 14895 | 271 ## and re-ordered function values after sort. |
| 272 | |
| 273 vbar = (sum (V(:,1:n)')/n)'; # Mean value | |
| 274 vr = (1 + alpha)*vbar - alpha*V(:,n+1); | |
| 275 x(:) = vr; | |
| 276 fr = dirn * feval (fun,x,varargin{:}); | |
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277 nf += 1; |
| 14895 | 278 vk = vr; fk = fr; how = "reflect, "; |
| 279 if (fr > f(n)) | |
| 280 if (fr > f(1)) | |
| 281 ve = gamma*vr + (1-gamma)*vbar; | |
| 282 x(:) = ve; | |
| 283 fe = dirn * feval (fun,x,varargin{:}); | |
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284 nf += 1; |
| 14895 | 285 if (fe > f(1)) |
| 286 vk = ve; | |
| 287 fk = fe; | |
| 288 how = "expand, "; | |
| 289 endif | |
| 290 endif | |
| 291 else | |
| 292 vt = V(:,n+1); | |
| 293 ft = f(n+1); | |
| 294 if (fr > ft) | |
| 295 vt = vr; | |
| 296 ft = fr; | |
| 297 endif | |
| 298 vc = beta*vt + (1-beta)*vbar; | |
| 299 x(:) = vc; | |
| 300 fc = dirn * feval (fun,x,varargin{:}); | |
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301 nf += 1; |
| 14895 | 302 if (fc > f(n)) |
| 303 vk = vc; fk = fc; | |
| 304 how = "contract,"; | |
| 305 else | |
| 306 for j = 2:n | |
| 307 V(:,j) = (V(:,1) + V(:,j))/2; | |
| 308 x(:) = V(:,j); | |
| 309 f(j) = dirn * feval (fun,x,varargin{:}); | |
| 310 endfor | |
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311 nf += n-1; |
| 14895 | 312 vk = (V(:,1) + V(:,n+1))/2; |
| 313 x(:) = vk; | |
| 314 fk = dirn * feval (fun,x,varargin{:}); | |
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315 nf += 1; |
| 14895 | 316 how = "shrink, "; |
| 317 endif | |
| 318 endif | |
| 319 V(:,n+1) = vk; | |
| 320 f(n+1) = fk; | |
| 321 [~,j] = sort(f); | |
| 322 j = j(n+1:-1:1); | |
| 323 f = f(j); | |
| 324 V = V(:,j); | |
| 325 | |
| 326 endwhile # End of outer (and only) loop. | |
| 327 | |
| 328 ## Finished. | |
| 329 if (trace) | |
| 330 fprintf (msg); | |
| 331 endif | |
| 332 x(:) = V(:,1); | |
| 333 | |
| 334 endfunction | |
| 335 | |
| 336 ## A helper function that evaluates a function and checks for bad results. | |
| 337 function y = guarded_eval (fun, x) | |
| 338 | |
| 339 y = fun (x); | |
| 340 | |
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341 if (! (isreal (y))) |
| 14895 | 342 error ("fminsearch:notreal", "fminsearch: non-real value encountered"); |
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343 elseif (any (isnan (y(:)))) |
| 14895 | 344 error ("fminsearch:isnan", "fminsearch: NaN value encountered"); |
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345 elseif (any (isinf (y(:)))) |
| 14895 | 346 error ("fminsearch:isinf", "fminsearch: Inf value encountered"); |
| 347 endif | |
| 348 | |
| 349 endfunction | |
| 350 | |
| 351 | |
| 352 %!demo | |
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353 %! fcn = @(x) (x(1)-5).^2 + (x(2)-8).^4; |
| 14895 | 354 %! x0 = [0;0]; |
| 355 %! [xmin, fval] = fminsearch (fcn, x0) | |
| 356 | |
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357 %!assert (fminsearch (@sin, 3, optimset ("MaxIter", 30)), 3*pi/2, 1e-4) |
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358 %!test |
| 14895 | 359 %! c = 1.5; |
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360 %! assert (fminsearch (@(x) x(1).^2+c*x(2).^2,[1;1]), [0;0], 1e-4); |
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361 |
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362 ## Additional argument |
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363 %!test |
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364 %! x1 = fminsearch (@(x, c) x(1).^2 + c*x(2).^2, [1;1], [], 1.5); |
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365 %! assert (x1, [0;0], 1e-4); |
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366 %! x1 = fminsearch (@(x, c) x(1).^2 + c*x(2).^2, [1;1], ... |
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367 %! optimset ("Display", "none"), 1.5); |
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368 %! assert (x1, [0;0], 1e-4); |
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369 |
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370 ## Test input validation |
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371 %!error fminsearch () |
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372 %!error fminsearch (1) |
| 14895 | 373 |
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374 ## Tests for guarded_eval |
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375 %!error <non-real value encountered> |
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376 %! fminsearch (@(x) ([0 2i]), 0, optimset ("FunValCheck", "on")); |
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377 %!error <NaN value encountered> |
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378 %! fminsearch (@(x) (NaN), 0, optimset ("FunValCheck", "on")); |
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379 %!error <Inf value encountered> |
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380 %! fminsearch (@(x) (Inf), 0, optimset ("FunValCheck", "on")); |