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function [xc,histout,costdata] = cgtrust(x0,f,parms,resolution)
%
%
% C. T. Kelley, Jan 13, 1998
%
% This code comes with no guarantee or warranty of any kind.
%
% function [x,histout] = cgtrust(x0,f,parms,resolution)
%
% Steihaug Newton-CG-Trust region algoirithm
%
% Input: x0 = initial iterate
% f = objective function,
% the calling sequence for f should be
% [fout,gout]=f(x) where fout=f(x) is a scalar
% and gout = grad f(x) is a COLUMN vector
% parms = [tol, eta, maxitnl, maxitl]
% tol = termination criterion norm(grad) < tol
% eta = forcing term in linear iteration (optional) default = .1
% maxitnl = maximum nonlinear iterations (optional) default = 100
% maxitl = maximum linear iterations (optional) default = 20
% resolution = estimated accuracy in functions/gradients (optional)
% default = 1.d-12
% The finite difference increment in the difference
% Hessian is set to sqrt(resolution).
%
%
%
% Output: x = solution
% histout = iteration history
% Each row of histout is
% [norm(grad), f, TR radius, inner iteration count]
% costdata = [num f, num grad]
%
% Requires: dirdero.m
%
% trust region algorithm parameters
%
omegaup=2; omegadown=.5;
mu0=.25; mulow=.25; muhigh=.75;
%
% set maxit, the resolution, and the difference increment
%
debug=0;
tol=parms(1); np=length(parms); itc=1;
if np < 2; eta=.1; else eta = parms(2); end
if np < 3; maxit=100; else maxit = parms(3); end
if np < 4; maxitl=20; else maxitl = parms(4); end
if nargin < 5
resolution = 1.d-12;
end
numf=0; numg=0;
hdiff=sqrt(resolution);
t=100; itc=0; xc=x0; n=length(x0);
[fc,gc]=feval(f,xc); numf=1; numg=1;
trrad=norm(x0);
ithist=zeros(maxit,4); ithist(itc+1,=[norm(gc), fc, trrad, 0];
%
paramstr=[eta,maxitl]; trcount=1;
while(norm(gc) > tol & itc <=maxit) & trcount < 30
itc=itc+1;
[s,dirs]=trcg(xc, f, gc, trrad, paramstr);
vd=size(dirs); itsl=vd(2); numf=numf+itsl; numg=numg+itsl;
xt=xc+s; ft=feval(f,xt); numf=numf+1; ared=ft-fc;
w= dirdero(xc, s, f, gc); numg=numg+1;
pred=gc'*s + .5*(w'*s); rat=ared/pred;
if rat > mu0
xc=xc+s; [fc,gc]=feval(f,xc); numf=numf+1;
if rat > muhigh & norm(s) > trrad-1.d-8 trrad=omegaup*trrad; end
if rat < mulow trrad=omegadown*trrad; end
else
for k=1:itsl dsums(k)=norm(sum(dirs(:,1:k),2)); end
trcount=1;
while rat <= mu0 & trcount < 30
trrad=omegadown*min(trrad,norm(s));
[s, kout]=tradj(trrad, dirs, itsl);
xt=xc+s; ft=feval(f,xt); numf=numf+1; ared=ft-fc;
%
% Only compute a new pred if ared < 0 and there's some hope
% that rat > mu0
%
if ared < 0
w= dirdero(xc, s, f, gc); numg=numg+1; pred=gc'*s + .5*(w'*s);
end
rat=ared/pred;
itsl=kout; trcount=trcount+1;
if trcount > 30
% ithist(itc+1,
% histout=ithist(1:itc+1,
disp(' stagnation in CGTR')
end
end
if trcount < 30
xc=xt; [fc,gc]=feval(f,xc); numf=numf+1; numg=numg+1;
end
end
ithist(itc+1,
end
histout=ithist(1:itc+1,
%
% find the point of intersetion of the TR boundary and the PL path
%
function [st, kout] = tradj(trrad, dirs, itsl)
st=dirs(:,1); inside=1;
if norm(st) > trrad | itsl == 1
st=st*trrad/norm(st);
kout=1;
else
for k=2:itsl
if norm(st+dirs(:,k)) > trrad & inside == 1
kout=k;
p=dirs(:,k); ac=p'*p; bc=2*(st'*p); cc=st'*st - trrad*trrad;
alpha=(-bc + sqrt(bc*bc - 4*ac*cc))/(2*ac); st=st+alpha*p;
inside=0;
else
st=st+dirs(:,k);
end
end
end
%
%
%
function [x, directions] = trcg(xc, f, gc, delta, params, pcv)
%
% Solve the trust region problem with preconditioned conjugate-gradient
%
% C. T. Kelley, January 13, 1997
%
% This code comes with no guarantee or warranty of any kind.
% function [x, directions]
% = trcg(xc, f, gc, delta)
%
%
%
% Input: xc=current point
% b=right hand side
% f = function, the calling sequence is
% [fun,grad]=f(x)
% gc = current gradient
% gc has usually been computed
% before the call to dirdero
% delta = TR radius
% params = two dimensional vector to control iteration
% params(1) = relative residual reduction factor
% params(2) = max number of iterations
% pcv, a routine to apply the preconditioner
% if omitted, the identity is used.
% The format for pcv is
% function px = pcv(x).
%
% Output: x = trial step
% directions = array of search directions TR radius reduction
%
%
%
% initialization
%
n=length(xc); errtol = params(1); maxiters = params(2);
x=zeros(n,1); b=-gc; r=b - dirdero(xc, x, f, gc);
if nargin == 5
z=r;
else
z = feval(pcv, r);
end
rho=z'*r;
tst=norm(r);
terminate=errtol*norm(b);
it=1;
directions=zeros(n,1);
hatdel=delta*(1-1.d-6);
while((tst > terminate) & (it <= maxiters) & norm(x) <= hatdel)
%
%
%
if(it==1)
p = z;
else
beta=rho/rhoold;
p = z + beta*p;
%
% end if
%
end
w = dirdero(xc, p, f, gc);
alpha=p'*w;
%
% If alpha <=0 head to the TR boundary and return
%
ineg=0;
if(alpha <= 0)
ac=p'*p; bc=2*(x'*p); cc=x'*x - delta*delta;
alpha=(-bc + sqrt(bc*bc - 4*ac*cc))/(2*ac);
disp(' negative curvature')
else
alpha=rho/alpha;
if norm(x+alpha*p) > delta
ac=p'*p; bc=2*(x'*p); cc=x'*x - delta*delta;
alpha=(-bc + sqrt(bc*bc - 4*ac*cc))/(2*ac);
end
end
x=x+alpha*p;
directions(:,it)=alpha*p;
r = r - alpha*w;
tst=norm(r);
rhoold=rho;
if nargin < 6 z=r; else z = feval(pcv, r); end
rho=z'*r;
it=it+1;
%
% end while
%
end
Originally Posted by thanhloc192 
