/usr/share/octave/packages/optim-1.4.1/private/__lm_feasible__.m is in octave-optim 1.4.1-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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##
## 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_res, objf, cvg, outp] = __lm_feasible__ (f, pin, hook)
## some backend specific defaults
fract_prec_default = 0;
max_fract_step_default = Inf;
n = length (pin);
## 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 function for gradient of objective function
grad_f = hook.dfdp;
## passed function for hessian of objective function
if (isempty (hessian = hook.hessian))
user_hessian = false;
A = eye (n);
else
user_hessian = true;
if (hook.inverse_hessian)
error ("this backend can't handle inverse hessians");
endif
endif
## passed function for complementary pivoting
cpiv = hook.cpiv;
## passed options
ftol = hook.TolFun;
if (isempty (niter = hook.MaxIter)) niter = 20; endif
fixed = hook.fixed;
maxstep = hook.max_fract_change;
maxstep(isna (maxstep)) = max_fract_step_default;
pprec = hook.fract_prec;
pprec(isna (pprec)) = fract_prec_default;
## keep absolute precision positive for non-null relative precision;
## arbitrary value, added to parameters before multiplying with
## relative precision
add_pprec = zeros (n, 1);
add_pprec(pprec > 0) = sqrt (eps);
##
verbose = strcmp (hook.Display, "iter");
user_interaction = hook.user_interaction;
## 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);
ac_idx = true (n_lcstr + n_gencstr, 1); # index of 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.
nz = 20 * eps; # This is arbitrary. Accuracy of equality constraints.
## backend-specific checking of options and constraints
##
if (any (pin < lbound | pin > ubound) ||
any (pin_cstr.inequ.lin_except_bounds < 0) ||
any (pin_cstr.inequ.gen < 0) ||
any (abs (pin_cstr.equ.lin) >= nz) ||
any (abs (pin_cstr.equ.gen) >= nz))
error ("Initial parameters violate constraints.");
endif
##
idx = lbound == ubound;
if (any (idx))
warning ("lower and upper bounds identical for some parameters, fixing the respective parameters");
fixed(idx) = true;
endif
if (all (fixed))
error ("no free parameters");
endif
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");
endif
## 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 = pbest = pin;
vf = fbest = f (pin);
iter = 0;
nobjf = 1;
done = false;
ll = 1;
ltab = [.1, 1, 1e2, 1e4, 1e6];
chgprev = Inf (n, 1);
df = [];
c_act = false (n, 1);
dca = zeros (n, 0);
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, p,
struct ("iteration", 0,
"fval", vf),
"init")))
p_res = p;
outp.niter = 0;
objf = vf;
cvg = -1;
return;
endif
while (! done)
iter++;
## gradient of objective function
old_df = df;
df = grad_f (p, setfield (dfdp_hook, "f", vf))(:);
## constraints, preparation of some constants
v_cstr = f_cstr (p, ac_idx);
old_c_act = c_act;
old_dca = dca;
c_act = v_cstr < nz | eq_idx; # index of active constraints
if (any (c_act))
if (n_gencstr)
## 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?
endif
dca = dcat.';
a_eq_idx = eq_idx(c_act);
else
dca = zeros (n, 0);
endif
## hessian of objectiv function
if (user_hessian)
A = hessian (p);
idx = isnan (A);
A(idx) = A.'(idx);
if (any (isnan (A(:))))
error ("some second derivatives undefined by user function");
endif
if (! isreal (A))
error ("second derivatives given by user function not real");
endif
if (! issymmetric (A))
error ("Hessian returned by user function not symmetric");
endif
elseif (iter > 1)
if (any (chg))
## approximate Hessian of Lagrangian
## I wonder if this hassle here and above with accounting for
## changing active sets is indeed better than just approximating
## the Hessian only of the objective function.
##
## index, over all constraints, of constraints active both
## previously and currently
s_c_act = old_c_act & c_act;
## index, over currently active constraints, of constraints
## active both previously and currently
id_new = s_c_act(c_act);
## index, over previously active constraints, of constraints
## active both previously and currently
id_old = s_c_act(old_c_act);
## gradients of currently active constraints which were also
## active previously
dca_new_id = dca(:, id_new);
## gradients of previously active constraints which are also
## active currently
dca_old_id = old_dca(:, id_old);
## index, over constraints active both previously and currently,
## of (old) non-zero multipliers (bidx set below previously)
bidx_old_id = bidx(id_old);
## index, over (old) non-zero multipliers, of constraints active
## both previously and currently (bidx set below previously)
old_l_idx = id_old(bidx);
## difference of derivatives of new and old active constraints,
## multiplied by multipliers, as used for BFGS update (lb set
## below previously)
dch = (dca_new_id(:, bidx_old_id) - ...
dca_old_id(:, bidx_old_id)) * ...
lb(old_l_idx);
y = df - old_df - dch;
## Damped BFGS according to Nocedal & Wright, 2nd edition,
## procedure 18.2.
chgt = chg.';
sAs = chgt * A * chg;
cy = chgt * y;
if (cy >= .2 * sAs)
th = 1;
else
if ((den1 = sAs - cy) == 0)
cvg = -4;
break;
endif
th = .8 * sAs / den1;
endif
Ac = A * chg;
r = th * y + (1 - th) * Ac;
if ((den2 = chgt * r) == 0 || sAs == 0)
cvg = -4;
break;
endif
A += r * r.' / den2 - Ac * Ac.' / sAs;
endif
endif
## Inverse scaled decomposition A = G * (1 ./ L) * G.'
##
## make matrix Binv for scaling
Binv = diag (A);
nidx = ! (idx = Binv == 0);
Binv(nidx) = 1 ./ sqrt (abs (Binv(nidx)));
Binv(idx) = 1;
Binv = diag (Binv);
## eigendecomposition of scaled A
[V, L] = eig (Binv * A * Binv);
L = diag (L);
## A is symmetric, so V and L are real, delete any imaginary parts,
## which might occur due to inaccuracy
V = real (V);
L = real (L);
##
nminL = - min (L) * 1.1 / ltab(1);
G = Binv * V;
## Levenberg/Marquardt
fgoal = vf - (abs (vf) + sqrt (eps)) * ftol;
for l = ltab
ll = max (ll, nminL);
l = max (1e-7, ll * l);
R = G * diag (1 ./ (L + l)) * G.';
## step computation
if (any (c_act))
## some constraints are active, quadratic programming
tp = dcat * R;
[lb, bidx, ridx, tbl] = cpiv (- tp * df, tp * dca, a_eq_idx);
chg = R * (dca(:, bidx) * lb - df); # step direction
## indices for different types of constraints
c_inact = ! c_act; # inactive constraints
c_binding = c_unbinding = nc_idx;
c_binding(c_act) = bidx; # constraints selected binding
c_unbinding(c_act) = ridx; # constraints unselected binding
c_nonbinding = c_act & ! (c_binding | c_unbinding); #
#constraints selected non-binding
else
## no constraints are active, chg is the Levenberg/Marquardt step
chg = - R * df; # step direction
lb = zeros (0, 1);
bidx = false (0, 1);
## indices for different types of constraints (meaning see above)
c_inact = ac_idx;
c_binding = nc_idx;
c_unbinding = nc_idx;
c_nonbinding = nc_idx;
endif
## apply inactive and non-binding constraints to step width
##
## linear constraints
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, :) * p) ./ ...
hstep(idx)));
endif
##
## general constraints
if (n_gencstr)
c_tp = gc_idx & (c_nonbinding | c_inact);
if (any (c_tp) && any (f_cstr (p + k * chg, c_tp) < 0))
[k, fval, info] = ...
fzero (@ (x) min (cat (1, ...
f_cstr (p + 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");
endif
endif
endif
##
chg = k * chg;
## if necessary, regain binding constraints and one of the
## possibly active previously inactive or non-binding constraints
if (any (gc_idx & c_binding)) # none selected binding => none
# unselected binding
ptp1 = p + chg;
tp = true;
nt_nosuc = true;
lim = 20;
while (nt_nosuc && lim >= 0)
## we keep d_p.' * inv (R) * d_p minimal in each step of the
## inner loop
c_tp0 = c_inact | c_nonbinding;
c_tp1 = c_inact | (gc_idx & c_nonbinding);
btbl = tbl(bidx, bidx);
c_tp2 = c_binding;
## once (any(tp)==false), it would not get true again even
## with the following assignment
if (any (tp) && ...
any (tp = f_cstr (ptp1, c_tp1) < nz))
## 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, :) * R * cadd;
Ct = C.';
T = [btbl, btbl * C; ...
-Ct * btbl, caddt * R * cadd - Ct * btbl * C];
btbl = gjp (T, size (T, 1));
endif
dcbt = dct(c_tp2, :);
mfc = - R * dcbt.' * btbl;
ptp2 = ptp1;
nt_niter = nt_niter_start = 100;
while (nt_nosuc && nt_niter >= 0)
hv = f_cstr (ptp2, c_tp2);
if (all (abs (hv) < nz))
nt_nosuc = false;
chg = ptp2 - p;
else
ptp2 = ptp2 + mfc * hv; # step should be zero for each
# component for which the parameter is
# "fixed"
endif
nt_niter--;
endwhile
if (nt_nosuc || ...
any (abs (chg) > abs (p .* maxstep)) || ...
any (f_cstr (ptp2, c_tp0) < -nz))
## if (nt_nosuc), regaining did not converge, else,
## regaining violated type 3 and 4.
nt_nosuc = true;
ptp1 = (p + ptp1) / 2;
endif
if (! nt_nosuc && ...
any ((tp = f_cstr (ptp2, c_unbinding)) < 0))
[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;
endif
endif
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 = (p + 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;
endif
## regaining violated type 2 constraints
nt_nosuc = true;
endif
lim--;
endwhile
if (nt_nosuc)
error ("could not regain binding constraints");
endif
else
## check the maximal stepwidth and apply as necessary
ochg = chg;
idx = ! isinf (maxstep);
limit = abs (maxstep(idx) .* p(idx));
chg(idx) = min (max (chg(idx), - limit), limit);
if (verbose && any (ochg != chg))
printf ("Change in parameter(s): %s:maximal fractional stepwidth enforced", ...
sprintf ("%d ", find (ochg != chg)));
endif
endif # regaining
aprec = pprec .* (abs (pbest) + add_pprec);
if (any (abs (chg) > 0.1 * aprec)) # only worth evaluating
# function if there is some
# non-miniscule change
skipped = false;
p_chg = p + chg;
## 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_chg < lbound;
uidx = p_chg > ubound;
p_chg(lidx, 1) = lbound(lidx, 1);
p_chg(uidx, 1) = ubound(uidx, 1);
chg(lidx, 1) = p_chg(lidx, 1) - p(lidx, 1);
chg(uidx, 1) = p_chg(uidx, 1) - p(uidx, 1);
endif
##
if (! isreal (vf_chg = f (p_chg)))
error ("objective function not real");
endif
nobjf++;
if (vf_chg < fbest)
pbest = p_chg;
fbest = vf_chg;
endif
if (vf_chg < fgoal) # <, not <=, since fgoal can be equal to vf
# if TolFun <= eps
p = p_chg;
vf = vf_chg;
break;
endif
else
skipped = true;
break;
endif
endfor
ll = l;
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, p,
struct ("iteration", iter,
"fval", vf),
"iter")))
p_res = p;
outp.niter = iter;
objf = vf;
cvg = -1;
return;
endif
aprec = pprec .* (abs (pbest) + add_pprec);
if (skipped)
cvg = 2;
done = true;
elseif (vf_chg >= fgoal) # >=, not >, since fgoal can be equal to vf
# if TolFun <= eps
cvg = 3;
done = true;
elseif (all (abs (chg) <= aprec) && all (abs (chgprev) <= aprec))
cvg = 2;
done = true;
elseif (iter == niter)
cvg = 0;
done = true;
else
chgprev = chg;
endif
endwhile
## return result
p_res = pbest;
objf = fbest;
outp.niter = iter;
outp.nobjf = nobjf;
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, p_res,
struct ("iteration", iter,
"fval", objf),
"done")))
cvg = -1;
endif
endfunction
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