Hi Gilad
Sorry I posted the incorrect calling arguments initially. I'm glad
everything work out in the end.
The algorithm, formally, is as follows:
function [X, info, perf, D] = ucminf(fun,par, x0, opts, D0)
%UCMINF BFGS method for unconstrained nonlinear optimization:
% Find xm = argmin{f(x)} , where x is an n-vector and the scalar
% function F with gradient g (with elements g(i) = DF/Dx_i )
% must be given by a MATLAB function with with declaration
% function [F, g] = fun(x, par)
% par holds parameters of the function. It may be dummy.
%
% Call: [X, info {, perf {, D}}] = ucminf(fun,par, x0, opts {,D0})
%
% Input parameters
% fun : String with the name of the function.
% par : Parameters of the function. May be empty.
% x0 : Starting guess for x .
% opts : Vector with 4 elements:
% opts(1) : Expected length of initial step
% opts(2:4) used in stopping criteria:
% ||g||_inf <= opts(2) or
% ||dx||_2 <= opts(3)*(opts(3) + ||x||_2) or
% no. of function evaluations exceeds opts(4) .
% Any illegal element in opts is replaced by its
% default value, [1 1e-4*||g(x0)||_inf 1e-8 100]
% D0 : (optional) If present, then approximate inverse Hessian
% at x0 . Otherwise, D0 := I
% Output parameters
% X : If perf is present, then array, holding the iterates
% columnwise. Otherwise, computed solution vector.
% info : Performance information, vector with 6 elements:
% info(1:3) = final values of [f(x) ||g||_inf ||dx||_2]
% info(4:5) = no. of iteration steps and evaluations of (F,g)
% info(6) = 1 : Stopped by small gradient
% 2 : Stopped by small x-step
% 3 : Stopped by opts(4) .
% 4 : Stopped by zero step.
% perf : (optional). If present, then array, holding
% perf(1:2,:) = values of f(x) and ||g||_inf
% perf(3:5,:) = Line search info: values of
% alpha, phi'(alpha), no. fct. evals.
% perf(6,:) = trust region radius.
% D : (optional). If present, then array holding the
% approximate inverse Hessian at X(:,end) .
% Hans Bruun Nielsen, IMM, DTU. 00.12.18 / 02.01.22
% Check call
[x n F g] = check(fun,par,x0,opts);
if nargin > 4, D = checkD(n,D0); fst = 0;
else, D = eye(n); fst = 1; end
% Finish initialization
k = 1; kmax = opts(4); neval = 1; ng = norm(g,inf);
Delta = opts(1);
Trace = nargout > 2;
if Trace
X = x(:)*ones(1,kmax+1);
perf = [F; ng; zeros(3,1); Delta]*ones(1,kmax+1);
end
found = ng <= opts(2);
h = zeros(size(x)); nh = 0;
ngs = ng * ones(1,3);
while ~found
% Previous values
xp = x; gp = g; Fp = F; nx = norm(x);
ngs = [ngs(2:3) ng];
h = D*(-g(:)); nh = norm(h); red = 0;
if nh <= opts(3)*(opts(3) + nx), found = 2;
else
if fst | nh > Delta % Scale to ||h|| = Delta
h = (Delta / nh) * h; nh = Delta;
fst = 0; red = 1;
end
k = k+1;
% Line search
[al F g dval slrat] = softline(fun,par,x,F,g, h);
if al < 1 % Reduce Delta
Delta = .35 * Delta;
elseif red & (slrat > .7) % Increase Delta
Delta = 3*Delta;
end
% Update x, neval and ||g||
x = x + al*h; neval = neval + dval; ng = norm(g,inf);
if Trace
X(:,k) = x(:);
perf(:,k) = [F; ng; al; dot(h,g); dval; Delta]; end
h = x - xp; nh = norm(h);
if nh == 0,
found = 4;
else
y = g - gp; yh = dot(y,h);
if yh > sqrt(eps) * nh * norm(y)
% Update D
v = D*y(:); yv = dot(y,v);
a = (1 + yv/yh)/yh; w = (a/2)*h(:) - v/yh;
D = D + w*h' + h*w';
end % update D
% Check stopping criteria
thrx = opts(3)*(opts(3) + norm(x));
if ng <= opts(2), found = 1;
elseif nh <= thrx, found = 2;
elseif neval >= kmax, found = 3;
% elseif neval > 20 & ng > 1.1*max(ngs), found = 5;
else, Delta = max(Delta, 2*thrx); end
end
end % Nonzero h
end % iteration
% Set return values
if Trace
X = X(:,1:k); perf = perf(:,1:k);
else, X = x; end
info = [F ng nh k-1 neval found];
% ========== auxiliary functions =================================
function [x,n, F,g, opts] = check(fun,par,x0,opts0)
% Check function call
x = x0(:); sx = size(x); n = max(sx);
if (min(sx) > 1)
error('x0 should be a vector'), end
[F g] = feval(fun,x,par);
sf = size(F); sg = size(g);
if any(sf-1) | ~isreal(F)
error('F should be a real valued scalar'), end
if (min(sg) ~= 1) | (max(sg) ~= n)
error('g should be a vector of the same length as x'), end
so = size(opts0);
if (min(so) ~= 1) | (max(so) < 4) | any(~isreal(opts0(1:4)))
error('opts should be a real valued vector of length 4'), end
opts = opts0(1:4); opts = opts(:).';
i = find(opts <= 0);
if length(i) % Set default values
d = [1 1e-4*norm(g, inf) 1e-8 100];
opts(i) = d(i);
end
% ---------- end of check ---------------------------------------
function D = checkD(n,D0)
% Check given inverse Hessian
D = D0; sD = size(D);
if any(sD - n)
error(sprintf('D should be a square matrix of size %g',n)), end
% Check symmetry
dD = D - D'; ndD = norm(dD(:),inf);
if ndD > 10*eps*norm(D(:),inf)
error('The given D0 is not symmetric'), end
if ndD, D = (D + D')/2; end % Symmetrize
[R p] = chol(D);
if p
error('The given D0 is not positive definite'), end
function [alpha,fn,gn,neval,slrat] = ...
softline(fun,fpar, x,f,g, h)
% Soft line search: Find alpha = argmin_a{f(x+a*h)}
% Default return values
alpha = 0; fn = f; gn = g; neval = 0; slrat = 1;
n = length(x);
% Initial values
dfi0 = dot(h,gn); if dfi0 >= 0, return, end
fi0 = f; slope0 = .05*dfi0; slopethr = .995*dfi0;
dfia = dfi0; stop = 0; ok = 0; neval = 0; b = 1;
while ~stop
[fib g] = feval(fun,x+b*h,fpar); neval = neval + 1;
dfib = dot(g,h);
if b == 1, slrat = dfib/dfi0; end
if fib <= fi0 + slope0*b % New lower bound
if dfib > abs(slopethr), stop = 1;
else
alpha = b; fn = fib; gn = g; dfia = dfib;
ok = 1; slrat = dfib/dfi0;
if (neval < 5) & (b < 2) & (dfib < slopethr)
% Augment right hand end
b = 2*b;
else, stop = 1; end
end
else, stop = 1; end
end
stop = ok; xfd = [alpha fn dfia; b fib dfib; b fib dfib];
while ~stop
c = interpolate(xfd,n);
[fic g] = feval(fun, x+c*h, fpar); neval = neval+1;
xfd(3,:) = [c fic dot(g,h)];
if fic < fi0 + slope0*c % New lower bound
xfd(1,:) = xfd(3,:); ok = 1;
alpha = c; fn = fic; gn = g; slrat = xfd(3,3)/dfi0;
else, xfd(2,:) = xfd(3,:); ok = 0; end
% Check stopping criteria
ok = ok & abs(xfd(3,3)) <= abs(slopethr);
stop = ok | neval >= 5 | diff(xfd(1:2,1)) <= 0;
end % while
%------------ end of softline ------------------------------
function alpha = interpolate(xfd,n);
% Minimizer of parabola given by
% xfd(1:2,1:3) = [a fi(a) fi'(a); b fi(b) dummy]
a = xfd(1,1); b = xfd(2,1); d = b - a; dfia = xfd(1,3);
C = diff(xfd(1:2,2)) - d*dfia;
if C >= 5*n*eps*b % Minimizer exists
A = a - .5*dfia*(d^2/C);
d = 0.1*d; alpha = min(max(a+d, A), b-d);
else
alpha = (a+b)/2;
end
%------------ end of interpolate -------------------------- |
