--- /dev/null
+%% Copyright (C) 1992-1994 Richard Shrager
+%% Copyright (C) 1992-1994 Arthur Jutan
+%% Copyright (C) 1992-1994 Ray Muzic
+%% Copyright (C) 2010, 2011 Olaf Till <olaf.till@uni-jena.de>
+%%
+%% This program 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.
+%%
+%% This program 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
+%% this program; if not, see <http://www.gnu.org/licenses/>.
+
+function [p, resid, cvg, outp] = __lm_svd__ (F, pin, hook)
+
+ %% This is a backend for optimization. This code was originally
+ %% contained in leasqr.m, which is now a frontend.
+
+ %% some backend specific defaults
+ fract_prec_default = 0;
+ max_fract_step_default = Inf;
+
+ %% needed for some anonymous functions
+ if (exist ('ifelse') ~= 5)
+ ifelse = @ scalar_ifelse;
+ end
+
+ %% passed constraints
+ mc = hook.mc; % matrix of linear constraints
+ vc = hook.vc; % vector of linear constraints
+ f_cstr = hook.f_cstr; % function of all constraints
+ df_cstr = hook.df_cstr; % function of derivatives of all constraints
+ n_gencstr = hook.n_gencstr; % number of non-linear constraints
+ eq_idx = hook.eq_idx; % logical index of equality constraints in all
+ % constraints
+ lbound = hook.lbound; % bounds, subset of linear inequality
+ ubound = hook.ubound; % constraints in mc and vc
+
+ %% passed values of constraints for initial parameters
+ pin_cstr = hook.pin_cstr;
+
+ %% passed return value of F for initial parameters
+ f_pin = hook.f_pin;
+
+ %% passed derivative of residual function
+ dfdp = hook.dfdp;
+
+ %% passed function for complementary pivoting
+ cpiv = hook.cpiv;
+
+ %% passed options
+ maxstep = hook.max_fract_change;
+ maxstep(isna (maxstep)) = max_fract_step_default;
+ pprec = hook.fract_prec;
+ pprec(isna (pprec)) = fract_prec_default;
+ stol = hook.TolFun;
+ niter = hook.MaxIter;
+ if (isempty (niter)) niter = 20; end
+ wt = hook.weights;
+ fixed = hook.fixed;
+ verbose = strcmp (hook.Display, 'iter');
+
+ %% only preliminary, for testing
+ if (isfield (hook, 'testing'))
+ testing = hook.testing;
+ else
+ testing = false;
+ end
+ if (isfield (hook, 'new_s'))
+ new_s = hook.new_s;
+ else
+ new_s = false;
+ end
+
+ %% some useful variables derived from passed variables
+ n_lcstr = size (vc, 1);
+ have_constraints_except_bounds = ...
+ n_lcstr + n_gencstr > ...
+ sum (lbound ~= -Inf) + sum (ubound ~= Inf);
+ n = length (pin);
+ wtl = wt(:);
+
+ nz = 20 * eps; % This is arbitrary. Constraint function will be
+ % regarded as <= zero if less than nz.
+
+ %% backend-specific checking of options and constraints
+ if (have_constraints_except_bounds)
+ if (any (pin_cstr.inequ.lin_except_bounds < 0) || ...
+ (n_gencstr > 0 && any (pin_cstr.inequ.gen < 0)))
+ warning ('initial parameters violate inequality constraints');
+ end
+ if (any (abs (pin_cstr.equ.lin) >= nz) || ...
+ (n_gencstr > 0 && any (abs (pin_cstr.equ.gen) >= nz)))
+ warning ('initial parameters violate equality constraints');
+ end
+ end
+ idx = lbound == ubound;
+ if (any (idx))
+ warning ('lower and upper bounds identical for some parameters, fixing the respective parameters');
+ fixed(idx) = true;
+ end
+ if (all (fixed))
+ error ('no free parameters');
+ end
+ lidx = pin < lbound;
+ uidx = pin > ubound;
+ if (any (lidx | uidx) && have_constraints_except_bounds)
+ warning ('initial parameters outside bounds, not corrected since other constraints are given');
+ else
+ if (any (lidx))
+ warning ('some initial parameters set to lower bound');
+ pin(lidx, 1) = lbound(lidx, 1);
+ end
+ if (any (uidx))
+ warning ('some initial parameters set to upper bound');
+ pin(uidx, 1) = ubound(uidx, 1);
+ end
+ end
+ if (n_gencstr > 0 && any (~isinf (maxstep)))
+ warning ('setting both a maximum fractional step change of parameters and general constraints may result in inefficiency and failure');
+ end
+
+ %% fill constant fields of hook for derivative-functions; some fields
+ %% may be backend-specific
+ dfdp_hook.fixed = fixed; % this may be handled by the frontend, but
+ % the backend still may add to it
+
+ %% set up for iterations
+ %%
+ p = pin;
+ f = f_pin; fbest=f; pbest=p;
+ m = prod (size (f));
+ r = wt .* f;
+ r = r(:);
+ if (~isreal (r)) error ('weighted residuals are not real'); end
+ ss = r.' * r;
+ sbest=ss;
+ chgprev=Inf*ones(n,1);
+ cvg=0;
+ epsLlast=1;
+ epstab=[.1, 1, 1e2, 1e4, 1e6];
+ ac_idx = true (n_lcstr + n_gencstr, 1); % all constraints
+ nc_idx = false (n_lcstr + n_gencstr, 1); % none of all constraints
+ gc_idx = cat (1, false (n_lcstr, 1), true (n_gencstr, 1)); % gen. constr.
+ lc_idx = ~gc_idx;
+
+ %% do iterations
+ %%
+ for iter = 1:niter
+ deb_printf (testing, '\nstart outer iteration\n');
+ v_cstr = f_cstr (p, ac_idx);
+ %% index of active constraints
+ c_act = v_cstr < nz | eq_idx; # equality constraints might be
+ # violated at start
+ if (any (c_act))
+ if (n_gencstr > 0)
+ %% full gradient is needed later
+ dct = df_cstr (p, ac_idx, ...
+ setfield (dfdp_hook, 'f', v_cstr));
+ dct(:, fixed) = 0; % for user supplied dfdp; necessary?
+ dcat = dct(c_act, :);
+ else
+ dcat = df_cstr (p, c_act, ...
+ setfield (dfdp_hook, 'f', v_cstr));
+ dcat(:, fixed) = 0; % for user supplied dfdp; necessary?
+ end
+ dca = dcat.';
+ end
+ nrm = zeros (1, n);
+ pprev=pbest;
+ prt = dfdp (p, setfield (dfdp_hook, 'f', fbest(:)));
+ prt(:, fixed) = 0; % for user supplied dfdp; necessary?
+ r = wt .* -fbest;
+ r = r(:);
+ if (~isreal (r)) error ('weighted residuals are not real'); end
+ sprev=sbest;
+ sgoal=(1-stol)*sprev;
+ msk = ~fixed;
+ prt(:, msk) = prt(:, msk) .* wtl(:, ones (1, sum (msk)));
+ nrm(msk) = sumsq (prt(:, msk), 1);
+ msk = nrm > 0;
+ nrm(msk) = 1 ./ sqrt (nrm(msk));
+ prt = prt .* nrm(ones (1, m), :);
+ nrm = nrm.';
+ [prt,s,v]=svd(prt,0);
+ s=diag(s);
+ g = prt.' * r;
+ for jjj=1:length(epstab)
+ deb_printf (testing, '\nstart inner iteration\n');
+ epsL = max(epsLlast*epstab(jjj),1e-7);
+ %% printf ('epsL: %e\n', epsL); % for testing
+
+ %% Usage of this 'ser' later is equivalent to pre-multiplying the
+ %% gradient with a positive-definit matrix, but not with a
+ %% diagonal matrix, at epsL -> Inf; so there is a fallback to
+ %% gradient descent, but not in general to descent for each
+ %% gradient component. Using the commented-out 'ser' ((1 / (1 +
+ %% epsL^2)) * (1 ./ se + epsL * s)) would be equivalent to using
+ %% Marquardts diagonal of the Hessian-approximation for epsL ->
+ %% Inf, but currently this gives no advantages in tests, even with
+ %% constraints.
+%%% ser = 1 ./ sqrt((s.*s)+epsL);
+ se = sqrt ((s.*s) + epsL);
+ if (new_s)
+ %% for testing
+ ser = (1 / (1 + epsL^2)) * (1 ./ se + epsL * s);
+ else
+ ser = 1 ./ se;
+ end
+ tp1 = (v * (g .* ser)) .* nrm;
+ if (any (c_act))
+ deb_printf (testing, 'constraints are active:\n');
+ deb_printf (testing, '%i\n', c_act);
+ %% calculate chg by 'quadratic programming'
+ nrme= diag (nrm);
+ ser2 = diag (ser .* ser);
+ mfc1 = nrme * v * ser2 * v.' * nrme;
+ tp2 = mfc1 * dca;
+ a_eq_idx = eq_idx(c_act);
+ [lb, bidx, ridx, tbl] = cpiv (dcat * tp1, dcat * tp2, a_eq_idx);
+ chg = tp1 + tp2(:, bidx) * lb; % if a parameter is 'fixed',
+ % the respective component of chg should
+ % be zero too, even here (with active
+ % constraints)
+ deb_printf (testing, 'change:\n');
+ deb_printf (testing, '%e\n', chg);
+ deb_printf (testing, '\n');
+ %% indices for different types of constraints
+ c_inact = ~c_act; % inactive constraints
+ c_binding = nc_idx;
+ c_binding(c_act) = bidx; % constraints selected binding
+ c_unbinding = nc_idx;
+ c_unbinding(c_act) = ridx; % constraints unselected binding
+ c_nonbinding = c_act & ~(c_binding | c_unbinding); % constraints
+ % selected non-binding
+ else
+ %% chg is the Levenberg/Marquardt step
+ chg = tp1;
+ %% indices for different types of constraints
+ c_inact = ac_idx; % inactive constraints consist of all
+ % constraints
+ c_binding = nc_idx;
+ c_unbinding = nc_idx;
+ c_nonbinding = nc_idx;
+ end
+ %% apply constraints to step width (since this is a
+ %% Levenberg/Marquardt algorithm, no line-search is performed
+ %% here)
+ k = 1;
+ c_tp = c_inact(1:n_lcstr);
+ mcit = mc(:, c_tp).';
+ vci = vc(c_tp);
+ hstep = mcit * chg;
+ idx = hstep < 0;
+ if (any (idx))
+ k = min (1, min (- (vci(idx) + mcit(idx, :) * pprev) ./ ...
+ hstep(idx)));
+ end
+ if (k < 1)
+ deb_printf (testing, 'stepwidth: linear constraints\n');
+ end
+ if (n_gencstr > 0)
+ c_tp = gc_idx & (c_nonbinding | c_inact);
+ if (any (c_tp) && any (f_cstr (pprev + k * chg, c_tp) < 0))
+ [k, fval, info] = ...
+ fzero (@ (x) min (cat (1, ...
+ f_cstr (pprev + x * chg, c_tp), ...
+ k - x, ...
+ ifelse (x < 0, -Inf, Inf))), ...
+ 0);
+ if (info ~= 1 || abs (fval) >= nz)
+ error ('could not find stepwidth to satisfy inactive and non-binding general inequality constraints');
+ end
+ deb_printf (testing, 'general constraints limit stepwidth\n');
+ end
+ end
+ chg = k * chg;
+
+ if (any (gc_idx & c_binding)) % none selected binding =>
+ % none unselected binding
+ deb_printf (testing, 'general binding constraints must be regained:\n');
+ %% regain binding constraints and one of the possibly active
+ %% previously inactive or non-binding constraints
+ ptp1 = pprev + chg;
+
+ tp = true;
+ nt_nosuc = true;
+ lim = 20;
+ while (nt_nosuc && lim >= 0)
+ deb_printf (testing, 'starting from new value of p in regaining:\n');
+ deb_printf (testing, '%e\n', ptp1);
+ %% we keep d_p.' * inv (mfc1) * d_p minimal in each step of
+ %% the inner loop; this is both sensible (this metric
+ %% considers a guess of curvature of sum of squared residuals)
+ %% and convenient (we have useful matrices available for it)
+ c_tp0 = c_inact | c_nonbinding;
+ c_tp1 = c_inact | (gc_idx & c_nonbinding);
+ btbl = tbl(bidx, bidx);
+ c_tp2 = c_binding;
+ if (any (tp)) % if none before, does not get true again
+ tp = f_cstr (ptp1, c_tp1) < nz;
+ if (any (tp)) % could be less clumsy, but ml-compatibility..
+ %% keep only the first true entry in tp
+ tp(tp) = logical (cat (1, 1, zeros (sum (tp) - 1, 1)));
+ %% supplement binding index with one (the first) getting
+ %% binding in c_tp1
+ c_tp2(c_tp1) = tp;
+ %% gradient of this added constraint
+ caddt = dct(c_tp2 & ~c_binding, :);
+ cadd = caddt.';
+ C = dct(c_binding, :) * mfc1 * cadd;
+ Ct = C.';
+ G = [btbl, btbl * C; ...
+ -Ct * btbl, caddt * mfc1 * cadd - Ct * btbl * C];
+ btbl = gjp (G, size (G, 1));
+ end
+ end
+ dcbt = dct(c_tp2, :);
+ mfc = - mfc1 * dcbt.' * btbl;
+ deb_printf (testing, 'constraints to regain:\n');
+ deb_printf (testing, '%i\n', c_tp2);
+
+ ptp2 = ptp1;
+ nt_niter_start = 100;
+ nt_niter = nt_niter_start;
+ while (nt_nosuc && nt_niter >= 0)
+ hv = f_cstr (ptp2, c_tp2);
+ if (all (abs (hv) < nz))
+ nt_nosuc = false;
+ chg = ptp2 - pprev;
+ else
+ ptp2 = ptp2 + mfc * hv; % step should be zero for each
+ % component for which the parameter is
+ % 'fixed'
+ end
+ nt_niter = nt_niter - 1;
+ end
+ deb_printf (testing, 'constraints after regaining:\n');
+ deb_printf (testing, '%e\n', hv);
+ if (nt_nosuc || ...
+ any (abs (chg) > abs (pprev .* maxstep)) || ...
+ any (f_cstr (ptp2, c_tp0) < -nz))
+ if (nt_nosuc)
+ deb_printf (testing, 'regaining did not converge\n');
+ else
+ deb_printf (testing, 'regaining violated type 3 and 4\n');
+ end
+ nt_nosuc = true;
+ ptp1 = (pprev + ptp1) / 2;
+ end
+ if (~nt_nosuc)
+ tp = f_cstr (ptp2, c_unbinding);
+ if (any (tp) < 0) % again ml-compatibility clumsyness..
+ [discarded, id] = min(tp);
+ tid = find (ridx);
+ id = tid(id); % index within active constraints
+ unsuccessful_exchange = false;
+ if (abs (tbl(id, id)) < nz) % Bard: not absolute value
+ %% exchange this unselected binding constraint against a
+ %% binding constraint, but not against an equality
+ %% constraint
+ tbidx = bidx & ~a_eq_idx;
+ if (~any (tbidx))
+ unsuccessful_exchange = true;
+ else
+ [discarded, idm] = max (abs (tbl(tbidx, id)));
+ tid = find (tbidx);
+ idm = tid(idm); % -> index within active constraints
+ tbl = gjp (tbl, idm);
+ bidx(idm) = false;
+ ridx(idm) = true;
+ end
+ end
+ if (unsuccessful_exchange)
+ %% It probably doesn't look good now; this desperate
+ %% last attempt is not in the original algortithm, since
+ %% that didn't account for equality constraints.
+ ptp1 = (pprev + ptp1) / 2;
+ else
+ tbl = gjp (tbl, id);
+ bidx(id) = true;
+ ridx(id) = false;
+ c_binding = nc_idx;
+ c_binding(c_act) = bidx;
+ c_unbinding = nc_idx;
+ c_unbinding(c_act) = ridx;
+ end
+ nt_nosuc = true;
+ deb_printf (testing, 'regaining violated type 2\n');
+ end
+ end
+ if (~nt_nosuc)
+ deb_printf (testing, 'regaining successful, converged with %i iterations:\n', ...
+ nt_niter_start - nt_niter);
+ deb_printf (testing, '%e\n', ptp2);
+ end
+ lim = lim - 1;
+ end
+ if (nt_nosuc)
+ error ('could not regain binding constraints');
+ end
+ else
+ %% check the maximal stepwidth and apply as necessary
+ ochg=chg;
+ idx = ~isinf(maxstep);
+ limit = abs(maxstep(idx).*pprev(idx));
+ chg(idx) = min(max(chg(idx),-limit),limit);
+ if (verbose && any(ochg ~= chg))
+ disp(['Change in parameter(s): ', ...
+ sprintf('%d ',find(ochg ~= chg)), 'maximal fractional stepwidth enforced']);
+ end
+ end
+ aprec=abs(pprec.*pbest); %---
+ %% ss=scalar sum of squares=sum((wt.*f)^2).
+ if (any(abs(chg) > 0.1*aprec))%--- % only worth evaluating
+ % function if there is some non-miniscule
+ % change
+ %% In the code of the outer loop before the inner loop pbest is
+ %% actually identical to p, since once they deviate, the outer
+ %% loop will not be repeated. Though the inner loop can still be
+ %% repeated in this case, pbest is not used in it. Since pprev
+ %% is set from pbest in the outer loop before the inner loop, it
+ %% is also identical to p up to here.
+ p=chg+pprev;
+ %% since the projection method may have slightly violated
+ %% constraints due to inaccuracy, correct parameters to bounds
+ %% --- but only if no further constraints are given, otherwise
+ %% the inaccuracy in honoring them might increase by this
+ if (~have_constraints_except_bounds)
+ lidx = p < lbound;
+ uidx = p > ubound;
+ p(lidx, 1) = lbound(lidx, 1);
+ p(uidx, 1) = ubound(uidx, 1);
+ chg(lidx, 1) = p(lidx, 1) - pprev(lidx, 1);
+ chg(uidx, 1) = p(uidx, 1) - pprev(uidx, 1);
+ end
+ %%
+ f = F (p);
+ r = wt .* f;
+ r = r(:);
+ if (~isreal (r))
+ error ('weighted residuals are not real');
+ end
+ ss = r.' * r;
+ deb_printf (testing, 'sbest: %.16e\n', sbest);
+ deb_printf (testing, 'sgoal: %.16e\n', sgoal);
+ deb_printf (testing, ' ss: %.16e\n', ss);
+ if (ss<sbest)
+ pbest=p;
+ fbest=f;
+ sbest=ss;
+ end
+ if (ss<=sgoal)
+ break;
+ end
+ end %---
+ end
+ %% printf ('epsL no.: %i\n', jjj); % for testing
+ epsLlast = epsL;
+ if (verbose)
+ hook.plot_cmd (f);
+ end
+ if (ss < eps) % in this case ss == sbest
+ cvg = 3; % there is no more suitable flag for this
+ break;
+ end
+ if (ss>sgoal)
+ cvg = 3;
+ break;
+ end
+ aprec=abs(pprec.*pbest);
+ %% [aprec, chg, chgprev]
+ if (all(abs(chg) <= aprec) && all(abs(chgprev) <= aprec))
+ cvg = 2;
+ if (verbose)
+ fprintf('Parameter changes converged to specified precision\n');
+ end
+ break;
+ else
+ chgprev=chg;
+ end
+ end
+
+ %% set further return values
+ %%
+ p = pbest;
+ resid = fbest;
+ outp.niter = iter;
+
+function deb_printf (do_printf, varargin)
+
+ %% for testing
+
+ if (do_printf)
+ printf (varargin{:})
+ end
+
+function fval = scalar_ifelse (cond, tval, fval)
+
+ %% needed for some anonymous functions, builtin ifelse only available
+ %% in Octave > 3.2; we need only the scalar case here
+
+ if (cond)
+ fval = tval;
+ end
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff