function [X, fX, i, fIter] = minimize(X, f, length, varargin);
% Minimize a differentiable multivariate function.
%
% Usage: [X, fX, i] = minimize(X, f, length, P1, P2, P3, ... )
%
% where the starting point is given by "X" (D by 1), and the function named in
% the string "f", must return a function value and a vector of partial
% derivatives of f wrt X, the "length" gives the length of the run: if it is
% positive, it gives the maximum number of line searches, if negative its
% absolute gives the maximum allowed number of function evaluations. You can
% (optionally) give "length" a second component, which will indicate the
% reduction in function value to be expected in the first line-search (defaults
% to 1.0). The parameters P1, P2, P3, ... are passed on to the function f.
%
% The function returns when either its length is up, or if no further progress
% can be made (ie, we are at a (local) minimum, or so close that due to
% numerical problems, we cannot get any closer). NOTE: If the function
% terminates within a few iterations, it could be an indication that the
% function values and derivatives are not consistent (ie, there may be a bug in
% the implementation of your "f" function). The function returns the found
% solution "X", a vector of function values "fX" indicating the progress made
% and "i" the number of iterations (line searches or function evaluations,
% depending on the sign of "length") used.
%
% The Polack-Ribiere flavour of conjugate gradients is used to compute search
% directions, and a line search using quadratic and cubic polynomial
% approximations and the Wolfe-Powell stopping criteria is used together with
% the slope ratio method for guessing initial step sizes. Additionally a bunch
% of checks are made to make sure that exploration is taking place and that
% extrapolation will not be unboundedly large.
%
% See also: checkgrad
%
% Copyright (C) 2001 - 2006 by Carl Edward Rasmussen (2006-02-23).
INT = 0.1; % don't reevaluate within 0.1 of the limit of the current bracket
EXT = 3.0; % extrapolate maximum 3 times the current step-size
MAX = 20; % max 20 function evaluations per line search
RATIO = 10; % maximum allowed slope ratio
SIG = 0.1; RHO = SIG/2; % SIG and RHO are the constants controlling the Wolfe-
% Powell conditions. SIG is the maximum allowed absolute ratio between
% previous and new slopes (derivatives in the search direction), thus setting
% SIG to low (positive) values forces higher precision in the line-searches.
% RHO is the minimum allowed fraction of the expected (from the slope at the
% initial point in the linesearch). Constants must satisfy 0 < RHO < SIG < 1.
% Tuning of SIG (depending on the nature of the function to be optimized) may
% speed up the minimization; it is probably not worth playing much with RHO.
% The code falls naturally into 3 parts, after the initial line search is
% started in the direction of steepest descent. 1) we first enter a while loop
% which uses point 1 (p1) and (p2) to compute an extrapolation (p3), until we
% have extrapolated far enough (Wolfe-Powell conditions). 2) if necessary, we
% enter the second loop which takes p2, p3 and p4 chooses the subinterval
% containing a (local) minimum, and interpolates it, unil an acceptable point
% is found (Wolfe-Powell conditions). Note, that points are always maintained
% in order p0 <= p1 <= p2 < p3 < p4. 3) compute a new search direction using
% conjugate gradients (Polack-Ribiere flavour), or revert to steepest if there
% was a problem in the previous line-search. Return the best value so far, if
% two consecutive line-searches fail, or whenever we run out of function
% evaluations or line-searches. During extrapolation, the "f" function may fail
% either with an error or returning Nan or Inf, and minimize should handle this
% gracefully.
if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
if length>0, S=['Linesearch']; else S=['Function evaluation']; end
i = 0; % zero the run length counter
ls_failed = 0; % no previous line search has failed
[f0 df0] = feval(f, X, varargin{:}); % get function value and gradient
fX = f0;
fIter = 0;
i = i + (length<0); % count epochs?!
s = -df0; d0 = -s'*s; % initial search direction (steepest) and slope
x3 = red/(1-d0); % initial step is red/(|s|+1)
while i < abs(length) % while not finished
i = i + (length>0); % count iterations?!
X0 = X; F0 = f0; dF0 = df0; % make a copy of current values
if length>0, M = MAX; else M = min(MAX, -length-i); end
while 1 % keep extrapolating as long as necessary
x2 = 0; f2 = f0; d2 = d0; df2 = df0; f3 = f0; df3 = df0;
success = 0;
while ~success & M > 0
try
M = M - 1; i = i + (length<0); % count epochs?!
[f3 df3] = feval(f, X+x3*s, varargin{:});
if isnan(f3) | isinf(f3) | any(isnan(df3)+isinf(df3)), error, end
success = 1;
catch % catch any error which occured in f
x3 = (x2+x3)/2; % bisect and try again
end
end
if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values
d3 = df3'*s; % new slope
if d3 > SIG*d0 | f3 > f0+x3*RHO*d0 | M == 0 % are we done extrapolating?
break
end
x1 = x2; f1 = f2; d1 = d2; df1 = df2; % move point 2 to point 1
x2 = x3; f2 = f3; d2 = d3; df2 = df3; % move point 3 to point 2
A = 6*(f1-f2)+3*(d2+d1)*(x2-x1); % make cubic extrapolation
B = 3*(f2-f1)-(2*d1+d2)*(x2-x1);
x3 = x1-d1*(x2-x1)^2/(B+sqrt(B*B-A*d1*(x2-x1))); % num. error possible, ok!
if ~isreal(x3) | isnan(x3) | isinf(x3) | x3 < 0 % num prob or wrong sign?
x3 = x2*EXT; % extrapolate maximum amount
elseif x3 > x2*EXT % new point beyond extrapolation limit?
x3 = x2*EXT; % extrapolate maximum amount
elseif x3 < x2+INT*(x2-x1) % new point too close to previous point?
x3 = x2+INT*(x2-x1);
end
end % end extrapolation
while (abs(d3) > -SIG*d0 | f3 > f0+x3*RHO*d0) & M > 0 % keep interpolating
if d3 > 0 | f3 > f0+x3*RHO*d0 % choose subinterval
x4 = x3; f4 = f3; d4 = d3; df4 = df3; % move point 3 to point 4
else
x2 = x3; f2 = f3; d2 = d3; df2 = df3; % move point 3 to point 2
end
if f4 > f0
x3 = x2-(0.5*d2*(x4-x2)^2)/(f4-f2-d2*(x4-x2)); % quadratic interpolation
else
A = 6*(f2-f4)/(x4-x2)+3*(d4+d2); % cubic interpolation
B = 3*(f4-f2)-(2*d2+d4)*(x4-x2);
x3 = x2+(sqrt(B*B-A*d2*(x4-x2)^2)-B)/A; % num. error possible, ok!
end
if isnan(x3) | isinf(x3)
x3 = (x2+x4)/2; % if we had a numerical problem then bisect
end
x3 = max(min(x3, x4-INT*(x4-x2)),x2+INT*(x4-x2)); % don't accept too close
[f3 df3] = feval(f, X+x3*s, varargin{:});
if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values
M = M - 1; i = i + (length<0); % count epochs?!
d3 = df3'*s; % new slope
end % end interpolation
if abs(d3) < -SIG*d0 & f3 < f0+x3*RHO*d0 % if line search succeeded
X = X+x3*s; f0 = f3; fX = [fX' f0]'; fIter = [fIter' i]'; % update variables
fprintf('%s %6i; Value %4.6e\n', S, i, f0);
s = (df3'*df3-df0'*df3)/(df0'*df0)*s - df3; % Polack-Ribiere CG direction
df0 = df3; % swap derivatives
d3 = d0; d0 = df0'*s;
if d0 > 0 % new slope must be negative
s = -df0; d0 = -s'*s; % otherwise use steepest direction
end
x3 = x3 * min(RATIO, d3/(d0-realmin)); % slope ratio but max RATIO
ls_failed = 0; % this line search did not fail
else
X = X0; f0 = F0; df0 = dF0; % restore best point so far
if ls_failed | i > abs(length) % line search failed twice in a row
break; % or we ran out of time, so we give up
end
s = -df0; d0 = -s'*s; % try steepest
x3 = 1/(1-d0);
ls_failed = 1; % this line search failed
end
end
fprintf('\n');