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author | Rik <rik@octave.org> |
---|---|
date | Wed, 28 Aug 2013 08:33:02 -0700 |
parents | bc924baa2c4e |
children | d63878346099 |
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## Copyright (C) 2008-2012 VZLU Prague, a.s. ## ## This file is part of Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## ## Author: Jaroslav Hajek <highegg@gmail.com> ## -*- texinfo -*- ## @deftypefn {Function File} {} fminunc (@var{fcn}, @var{x0}) ## @deftypefnx {Function File} {} fminunc (@var{fcn}, @var{x0}, @var{options}) ## @deftypefnx {Function File} {[@var{x}, @var{fvec}, @var{info}, @var{output}, @var{grad}, @var{hess}] =} fminunc (@var{fcn}, @dots{}) ## Solve an unconstrained optimization problem defined by the function ## @var{fcn}. ## ## @var{fcn} should accepts a vector (array) defining the unknown variables, ## and return the objective function value, optionally with gradient. ## In other words, this function attempts to determine a vector @var{x} such ## that @code{@var{fcn} (@var{x})} is a local minimum. ## @var{x0} determines a starting guess. The shape of @var{x0} is preserved ## in all calls to @var{fcn}, but otherwise is treated as a column vector. ## @var{options} is a structure specifying additional options. ## Currently, @code{fminunc} recognizes these options: ## @qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"}, ## @qcode{"TolFun"}, @qcode{"MaxIter"}, @qcode{"MaxFunEvals"}, ## @qcode{"GradObj"}, @qcode{"FinDiffType"}, ## @qcode{"TypicalX"}, @qcode{"AutoScaling"}. ## ## If @qcode{"GradObj"} is @qcode{"on"}, it specifies that @var{fcn}, ## called with 2 output arguments, also returns the Jacobian matrix ## of right-hand sides at the requested point. @qcode{"TolX"} specifies ## the termination tolerance in the unknown variables, while ## @qcode{"TolFun"} is a tolerance for equations. Default is @code{1e-7} ## for both @qcode{"TolX"} and @qcode{"TolFun"}. ## ## For description of the other options, see @code{optimset}. ## ## On return, @var{fval} contains the value of the function @var{fcn} ## evaluated at @var{x}, and @var{info} may be one of the following values: ## ## @table @asis ## @item 1 ## Converged to a solution point. Relative gradient error is less than ## specified ## by TolFun. ## ## @item 2 ## Last relative step size was less that TolX. ## ## @item 3 ## Last relative decrease in function value was less than TolF. ## ## @item 0 ## Iteration limit exceeded. ## ## @item -3 ## The trust region radius became excessively small. ## @end table ## ## Optionally, fminunc can also yield a structure with convergence statistics ## (@var{output}), the output gradient (@var{grad}) and approximate Hessian ## (@var{hess}). ## ## Notes: If you only have a single nonlinear equation of one variable then ## using @code{fminbnd} is usually a much better idea. The algorithm used is a ## gradient search which depends on the objective function being differentiable. ## If the function has discontinuities it may be better to use a derivative-free ## algorithm such as @code{fminsearch}. ## @seealso{fminbnd, fminsearch, optimset} ## @end deftypefn ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. ## PKG_ADD: [~] = __all_opts__ ("fminunc"); function [x, fval, info, output, grad, hess] = fminunc (fcn, x0, options = struct ()) ## Get default options if requested. if (nargin == 1 && ischar (fcn) && strcmp (fcn, 'defaults')) x = optimset ("MaxIter", 400, "MaxFunEvals", Inf, "GradObj", "off", "TolX", 1e-7, "TolFun", 1e-7, "OutputFcn", [], "FunValCheck", "off", "FinDiffType", "central", "TypicalX", [], "AutoScaling", "off"); return; endif if (nargin < 2 || nargin > 3 || ! ismatrix (x0)) print_usage (); endif if (ischar (fcn)) fcn = str2func (fcn, "global"); endif xsiz = size (x0); n = numel (x0); has_grad = strcmpi (optimget (options, "GradObj", "off"), "on"); cdif = strcmpi (optimget (options, "FinDiffType", "central"), "central"); maxiter = optimget (options, "MaxIter", 400); maxfev = optimget (options, "MaxFunEvals", Inf); outfcn = optimget (options, "OutputFcn"); ## Get scaling matrix using the TypicalX option. If set to "auto", the ## scaling matrix is estimated using the jacobian. typicalx = optimget (options, "TypicalX"); if (isempty (typicalx)) typicalx = ones (n, 1); endif autoscale = strcmpi (optimget (options, "AutoScaling", "off"), "on"); if (! autoscale) dg = 1 ./ typicalx; endif funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); if (funvalchk) ## Replace fcn with a guarded version. fcn = @(x) guarded_eval (fcn, x); endif ## These defaults are rather stringent. I think that normally, user ## prefers accuracy to performance. macheps = eps (class (x0)); tolx = optimget (options, "TolX", 1e-7); tolf = optimget (options, "TolFun", 1e-7); factor = 0.1; ## FIXME: TypicalX corresponds to user scaling (???) autodg = true; niter = 1; nfev = 0; x = x0(:); info = 0; ## Initial evaluation. fval = fcn (reshape (x, xsiz)); n = length (x); if (! isempty (outfcn)) optimvalues.iter = niter; optimvalues.funccount = nfev; optimvalues.fval = fval; optimvalues.searchdirection = zeros (n, 1); state = 'init'; stop = outfcn (x, optimvalues, state); if (stop) info = -1; break; endif endif nsuciter = 0; lastratio = 0; grad = []; ## Outer loop. while (niter < maxiter && nfev < maxfev && ! info) grad0 = grad; ## Calculate function value and gradient (possibly via FD). if (has_grad) [fval, grad] = fcn (reshape (x, xsiz)); grad = grad(:); nfev ++; else grad = __fdjac__ (fcn, reshape (x, xsiz), fval, typicalx, cdif)(:); nfev += (1 + cdif) * length (x); endif if (niter == 1) ## Initialize by identity matrix. hesr = eye (n); else ## Use the damped BFGS formula. y = grad - grad0; sBs = sumsq (w); Bs = hesr'*w; sy = y'*s; theta = 0.8 / max (1 - sy / sBs, 0.8); r = theta * y + (1-theta) * Bs; hesr = cholupdate (hesr, r / sqrt (s'*r), "+"); [hesr, info] = cholupdate (hesr, Bs / sqrt (sBs), "-"); if (info) hesr = eye (n); endif endif if (autoscale) ## Second derivatives approximate the hessian. d2f = norm (hesr, 'columns').'; if (niter == 1) dg = d2f; else ## FIXME: maybe fixed lower and upper bounds? dg = max (0.1*dg, d2f); endif endif if (niter == 1) xn = norm (dg .* x); ## FIXME: something better? delta = factor * max (xn, 1); endif ## FIXME -- why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector ## of perturbations of x, then norm (hesr*e) <= eps*xn, i.e. by ## tolf ~ eps we demand as much accuracy as we can expect. if (norm (grad) <= tolf*n*xn) info = 1; break; endif suc = false; decfac = 0.5; ## Inner loop. while (! suc && niter <= maxiter && nfev < maxfev && ! info) s = - __doglegm__ (hesr, grad, dg, delta); sn = norm (dg .* s); if (niter == 1) delta = min (delta, sn); endif fval1 = fcn (reshape (x + s, xsiz)) (:); nfev ++; if (fval1 < fval) ## Scaled actual reduction. actred = (fval - fval1) / (abs (fval1) + abs (fval)); else actred = -1; endif w = hesr*s; ## Scaled predicted reduction, and ratio. t = 1/2 * sumsq (w) + grad'*s; if (t < 0) prered = -t/(abs (fval) + abs (fval + t)); ratio = actred / prered; else prered = 0; ratio = 0; endif ## Update delta. if (ratio < min (max (0.1, 0.8*lastratio), 0.9)) delta *= decfac; decfac ^= 1.4142; if (delta <= 1e1*macheps*xn) ## Trust region became uselessly small. info = -3; break; endif else lastratio = ratio; decfac = 0.5; if (abs (1-ratio) <= 0.1) delta = 1.4142*sn; elseif (ratio >= 0.5) delta = max (delta, 1.4142*sn); endif endif if (ratio >= 1e-4) ## Successful iteration. x += s; xn = norm (dg .* x); fval = fval1; nsuciter ++; suc = true; endif niter ++; ## FIXME: should outputfcn be only called after a successful iteration? if (! isempty (outfcn)) optimvalues.iter = niter; optimvalues.funccount = nfev; optimvalues.fval = fval; optimvalues.searchdirection = s; state = 'iter'; stop = outfcn (x, optimvalues, state); if (stop) info = -1; break; endif endif ## Tests for termination conditions. A mysterious place, anything ## can happen if you change something here... ## The rule of thumb (which I'm not sure M*b is quite following) ## is that for a tolerance that depends on scaling, only 0 makes ## sense as a default value. But 0 usually means uselessly long ## iterations, so we need scaling-independent tolerances wherever ## possible. ## The following tests done only after successful step. if (ratio >= 1e-4) ## This one is classic. Note that we use scaled variables again, ## but compare to scaled step, so nothing bad. if (sn <= tolx*xn) info = 2; ## Again a classic one. elseif (actred < tolf) info = 3; endif endif endwhile endwhile ## Restore original shapes. x = reshape (x, xsiz); output.iterations = niter; output.successful = nsuciter; output.funcCount = nfev; if (nargout > 5) hess = hesr'*hesr; endif endfunction ## An assistant function that evaluates a function handle and checks for ## bad results. function [fx, gx] = guarded_eval (fun, x) if (nargout > 1) [fx, gx] = fun (x); else fx = fun (x); gx = []; endif if (! (isreal (fx) && isreal (gx))) error ("fminunc:notreal", "fminunc: non-real value encountered"); elseif (any (isnan (fx(:)))) error ("fminunc:isnan", "fminunc: NaN value encountered"); elseif (any (isinf (fx(:)))) error ("fminunc:isinf", "fminunc: Inf value encountered"); endif endfunction %!function f = __rosenb (x) %! n = length (x); %! f = sumsq (1 - x(1:n-1)) + 100 * sumsq (x(2:n) - x(1:n-1).^2); %!endfunction %!test %! [x, fval, info, out] = fminunc (@__rosenb, [5, -5]); %! tol = 2e-5; %! assert (info > 0); %! assert (x, ones (1, 2), tol); %! assert (fval, 0, tol); %!test %! [x, fval, info, out] = fminunc (@__rosenb, zeros (1, 4)); %! tol = 2e-5; %! assert (info > 0); %! assert (x, ones (1, 4), tol); %! assert (fval, 0, tol); %% Test FunValCheck works correctly %!assert (fminunc (@(x) x^2, 1, optimset ("FunValCheck", "on")), 0, eps) %!error <non-real value> fminunc (@(x) x + i, 1, optimset ("FunValCheck", "on")) %!error <NaN value> fminunc (@(x) x + NaN, 1, optimset ("FunValCheck", "on")) %!error <Inf value> fminunc (@(x) x + Inf, 1, optimset ("FunValCheck", "on")) ## Solve the double dogleg trust-region minimization problem: ## Minimize 1/2*norm(r*x)^2 subject to the constraint norm(d.*x) <= delta, ## x being a convex combination of the gauss-newton and scaled gradient. ## TODO: error checks ## TODO: handle singularity, or leave it up to mldivide? function x = __doglegm__ (r, g, d, delta) ## Get Gauss-Newton direction. b = r' \ g; x = r \ b; xn = norm (d .* x); if (xn > delta) ## GN is too big, get scaled gradient. s = g ./ d; sn = norm (s); if (sn > 0) ## Normalize and rescale. s = (s / sn) ./ d; ## Get the line minimizer in s direction. tn = norm (r*s); snm = (sn / tn) / tn; if (snm < delta) ## Get the dogleg path minimizer. bn = norm (b); dxn = delta/xn; snmd = snm/delta; t = (bn/sn) * (bn/xn) * snmd; t -= dxn * snmd^2 - sqrt ((t-dxn)^2 + (1-dxn^2)*(1-snmd^2)); alpha = dxn*(1-snmd^2) / t; else alpha = 0; endif else alpha = delta / xn; snm = 0; endif ## Form the appropriate convex combination. x = alpha * x + ((1-alpha) * min (snm, delta)) * s; endif endfunction