/usr/share/octave/packages/3.2/optim-1.0.17/private/__lm_svd__.m is in octave-optim 1.0.17-1.
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%% 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 2 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); % non 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)
dct = df_cstr (p, ac_idx, ...
setfield (dfdp_hook, 'f', v_cstr));
dct(:, fixed) = 0; % for user supplied dfdp; necessary?
dc = dct.';
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
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
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