--- /dev/null
+## Copyright (C) 2012 Olaf Till <i7tiol@t-online.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_res, objf, cvg, outp] = __lm_feasible__ (f, pin, hook)
+
+ ## 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;
+ endif
+
+ 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 return value of f for initial parameters
+ f_pin = hook.f_pin;
+
+ ## 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;
+ 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;
+ verbose = strcmp (hook.Display, "iter");
+
+ ## 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;
+ done = false;
+ ll = 1;
+ ltab = [.1, 1, 1e2, 1e4, 1e6];
+ chgprev = Inf (n, 1);
+ df = [];
+ c_act = false (n, 1);
+ dca = zeros (n, 0);
+
+ while (! done)
+
+ iter++;
+
+ ## gradient of objective function
+ old_df = df;
+ df = grad_f (p, setfield (dfdp_hook, "f", f))(:);
+
+ ## 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 = (1 - ftol) * vf;
+ 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 = abs (pprec .* pbest);
+ if (any (abs (chg) > 0.1 * aprec)) # only worth evaluating
+ # function if there is some
+ # non-miniscule change
+ 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
+ if (vf_chg < fbest)
+ pbest = p_chg;
+ fbest = vf_chg;
+ endif
+ if (vf_chg <= fgoal)
+ p = p_chg;
+ vf = vf_chg;
+ break;
+ endif
+ endif
+ endfor
+
+ ll = l;
+
+ aprec = abs (pprec .* pbest);
+ if (vf_chg < eps || vf_chg > fgoal)
+ 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;
+
+endfunction
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff