function [x,v,opt] = quadprog2(varargin)
% QUADPROG2 - Convex Quadratic Programming Solver
% Featuring the SOLVOPT freeware optimizer
%
% New for version 1.1: * Significant Speed Improvement
% * Geometric Preconditioning
% * Improved Error Checking
%
% USAGE:
% [x,v] = quadprog2(H,f,A,b)
% [x,v] = quadprog2(H,f,A,b,guess)
% [x,v,opt] = ...
%
% Minimizes the function v = 0.5*x'*H*x + f*x
% subject to the constraint A*x <= b.
% Initial guess is optional.
%
% ("opt" returns SOLVOPT data for advanced use. Details are available in
% the SOLVOPT documentation at the website identified below.)
%
% Notes: (1) For a problem with 100 variables and 300 constraints, you will
% often get a result in under 5 seconds. However, sometimes
% the optimizer has to work longer (see below) for difficult
% optimizations. Alerts are provided. (Note: The calculation
% time is more sensitive to the number of variables than it is
% to the number of contraints.)
% (2) Geometric preconditioning is undertaken for 10 or more
% dimensions to greatly reduce calculation time. (With fewer
% than 10 dimensions, there is negligible benefit, so the
% preconditioning calculations are omitted.)
% (3) Geometric preconditioning can impair the convergence of some
% difficult optimizations. When this occurs, the optimization
% is attempted again without the preconditioning.
% (4) x and guess are column vectors. f is a row vector.
% They will be converted if necessary.
% (5) This m-file incorporates the SOLVOPT version 1.1 freeware
% optimizer, which has been wholly reproduced, except for
% a few slight modifications for convenience in parameter passing.
% (6) SolvOpt is a general nonlinear local optimizer,
% written by Alexei Kuntsevich & Franz Kappel, and
% is available (as of this writing) as freeware from:
% http://www.uni-graz.at/imawww/kuntsevich/solvopt/
% (7) This Matlab function requires a convex QP problem
% with a positive-definite symmetric matrix H.
% This is a somewhat trivial application of
% a general solver like SOLVOPT, but the use of precomputed
% gradient vectors herein makes the solution fast enough
% to warrant use.
% (8) Any local solution of a convex QP is also a global solution.
% Hence, your results will be globally optimal.
% (9) Relative precision in the objective function is set to 1e-6.
% (10) Absolute precision in constraint violation is 1e-6 or better.
% (11) This program does not require the Optimization Toolbox
% (12) ver 1.0: intial writing, Michael Kleder, June 2005
% (13) ver 1.1: geometric preconditioning, Michael Kleder, July 2005
%
% EXAMPLE:
% % Convex QP with 100 variables and 300 constraints:
% n = 100;
% c = 300;
% H = rand(n);
% H=H*H';
% f=rand(1,n);
% A=rand(c,n)*2-1;
% b=ones(c,1);
% tic
% x = quadprog2(H,f,A,b)
% toc
[H,f,A,b]=deal(varargin{1:4});
if any(eig(H)<=0)
error('Non-convex. (Quadratic form is not positive definite.)')
end
if any(H~=H')
error('Quadratic form must be symmetric.')
end
f=f(:)';
n=size(H,2);
% try a quick check on unconstrained solution
HI=inv(H);
mqf=-HI*f'; % inverse exists since H is positive definite
if all(A*mqf<=b(:))
x=mqf;
v = .5*x'*H*x + f*x;
opt=[];
return
end
if nargin < 5
x=mqf+(rand(n,1)-.5);
else
x=varargin{5};
x=x(:);
end
if n>=10
% geometric pre-conditioning:
bn = b - A*mqf;
fn = 0*f;
T = sqrtm(HI);
An = A*T;
nA=sqrt(sum(An.^2,2));
An = An ./ repmat(nA,[1 size(An,2)]);
bn = bn ./ nA;
Hn = eye(size(H));
% invoke nonlinear solver:
pass={Hn,fn,An,bn};
%opt=[-1,1e-4,1e-6,15000,0,1e-8,2.5,1e-11]; % defaults
opt=[-1,1e-6,1e-8,15e5,0,1e-8,2.5,1e-11]; % increase precision
%opt=[-1,1e-14,1e-14,15e5,0,1e-14,2.5,1e-12]; % very high precision
%[x,v,opt] = solvopt2(x,@obj,[],opt,[],[],pass);
opt(6) = opt(6) * .1; % precondition needs constraint tolerance reduced
[x,v,opt] = solvopt2(x,@obj,@gobj,opt,@pen,@gpen,pass);
opt(6) = opt(6) * .1;
x = T*x+mqf;
v = .5*x'*H*x + f*x;
if max(A*x-b) <= 1e-6
return
else
disp('Difficult optimization. Omitting geometric preconditioning.')
pass={H,f,A,b};
[x,v,opt] = solvopt2(x,@obj,@gobj,opt,@pen,@gpen,pass);
if max(A*x-b) > opt(6)
disp('Feasible optimum not found. Searching again.')
x=mqf+(rand(n,1)-.5);
[x,v,opt] = solvopt2(x,@obj,@gobj,opt,@pen,@gpen,pass);
if max(A*x-b) > opt(6)
error('Optimization failure: Feasible optimum not found.')
end
end
return
end
else % low-dimensional
pass={H,f,A,b};
%opt=[-1,1e-4,1e-6,15000,0,1e-8,2.5,1e-11]; % defaults
opt=[-1,1e-6,1e-8,15e5,0,1e-8,2.5,1e-11]; % increase precision
%opt=[-1,1e-14,1e-14,15e5,0,1e-14,2.5,1e-12]; % very high precision
[x,v,opt] = solvopt2(x,@obj,@gobj,opt,@pen,@gpen,pass);
if max(A*x-b) > opt(6)
disp('Feasible optimum not found. Searching again.')
x=mqf+(rand(n,1)-.5);
[x,v,opt] = solvopt2(x,@obj,@gobj,opt,@pen,@gpen,pass);
if max(A*x-b) > opt(6)
error('Optimization failure: feasible optimum not found.')
end
end
end
return
function q = obj(x,pass) % objective function
H=pass{1};
f=pass{2};
q = .5*x'*H*x+f*x;
return
function q = gobj(x,pass); % gradient of objective function
H=pass{1};
f=pass{2};
q = H*x+f';
return
function q=pen(x,pass) % contraint penalty
A=pass{3};
b=pass{4};
q=max(0,A*x-b);
q=sum(q(:));
return
function q=gpen(x,pass) % gradient of contraint penalty
A=pass{3};
b=pass{4};
q=A*x-b;
q(q<0)=0;
q = q'*A/sqrt(q'*q);
return
function [x,f,options]=solvopt2(x,fun,grad,options,func,gradc,pass)
% added 'pass' input to pass params to fun/func/grad/gradc; also quieted warnings
% Usage:
% [x,f,options]=solvopt(x,fun,grad,options,func,gradc)
% The function SOLVOPT performs a modified version of Shor's r-algorithm in
% order to find a local minimum resp. maximum of a nonlinear function
% defined on the n-dimensional Euclidean space
% or
% a solution of a nonlinear constrained problem:
% min { f(x): g(x) (<)= 0, g(x) in R(m), x in R(n) }
% Arguments:
% x is the n-vector (row or column) of the coordinates of the starting
% point,
% fun is the name of an M-file (M-function) which computes the value
% of the objective function <fun> at a point x,
% synopsis: f=fun(x)
% grad is the name of an M-file (M-function) which computes the gradient
% vector of the function <fun> at a point x,
% synopsis: g=grad(x)
% func is the name of an M-file (M-function) which computes the MAXIMAL
% RESIDUAL(!) for a set of constraints at a point x,
% synopsis: fc=func(x)
% gradc is the name of an M-file (M-function) which computes the gradient
% vector for the maximal residual consyraint at a point x,
% synopsis: gc=gradc(x)
% options is a row vector of optional parameters:
% options(1)= H, where sign(H)=-1 resp. sign(H)=+1 means minimize
% resp. maximize <fun> (valid only for unconstrained problem)
% and H itself is a factor for the initial trial step size
% (options(1)=-1 by default),
% options(2)= relative error for the argument in terms of the
% infinity-norm (1.e-4 by default),
% options(3)= relative error for the function value (1.e-6 by default),
% options(4)= limit for the number of iterations (15000 by default),
% options(5)= control of the display of intermediate results and error
% resp. warning messages (default value is 0, i.e., no intermediate
% output but error and warning messages, see more in the manual),
% options(6)= admissible maximal residual for a set of constraints
% (options(6)=1e-8 by default, see more in the manual),
% *options(7)= the coefficient of space dilation (2.5 by default),
% *options(8)= lower bound for the stepsize used for the difference
% approximation of gradients (1e-12 by default, see more in the manual).
% (* ... changes should be done with care)
% Returned values:
% x is the optimizer (row resp. column),
% f is the optimum function value,
% options returns the values of the counters
% options(9), the number of iterations, if positive,
% or an abnormal stop code, if negative (see more in the manual),
% options(10), the number of objective
% options(11), the number of gradient evaluations,
% options(12), the number of constraint function evaluations,
% options(13), the number of constraint gradient evaluations.
% ____________________________________________________________________________
shhh = 1; % silence warnings (not errors)
% strings: ----{
errmes='SolvOpt error:';
wrnmes='SolvOpt warning:';
error1='No function name and/or starting point passed to the function.';
error2='Argument X has to be a row or column vector of dimension > 1.';
error30='<fun> returns an empty string.';
error31='Function value does not exist (NaN is returned).';
error32='Function equals infinity at the point.';
error40='<grad> returns an improper matrix. Check the dimension.';
error41='Gradient does not exist (NaN is returned by <grad>).';
error42='Gradient equals infinity at the starting point.';
error43='Gradient equals zero at the starting point.';
error50='<func> returns an empty string.';
error51='<func> returns NaN at the point.';
error52='<func> returns infinite value at the point.';
error60='<gradc> returns an improper vector. Check the dimension';
error61='<gradc> returns NaN at the point.';
error62='<gradc> returns infinite vector at the point.';
error63='<gradc> returns zero vector at an infeasible point.';
error5='Function is unbounded.';
error6='Choose another starting point.';
warn1= 'Gradient is zero at the point, but stopping criteria are not fulfilled.';
warn20='Normal re-setting of a transformation matrix.' ;
warn21='Re-setting due to the use of a new penalty coefficient.' ;
warn4= 'Iterations limit exceeded.';
warn31='The function is flat in certain directions.';
warn32='Trying to recover by shifting insensitive variables.';
warn09='Re-run from recorded point.';
warn08='Ravine with a flat bottom is detected.';
termwarn0='SolvOpt: Normal termination.';
termwarn1='SolvOpt: Termination warning:';
appwarn='The above warning may be reasoned by inaccurate gradient approximation';
endwarn=[...
'Premature stop is possible. Try to re-run the routine from the obtained point. ';...
'Result may not provide the optimum. The function apparently has many extremum points. ';...
'Result may be inaccurate in the coordinates. The function is flat at the optimum. ';...
'Result may be inaccurate in a function value. The function is extremely steep at the optimum.'];
% ----}
% ARGUMENTS PASSED ----{
if nargin<2 % Function and/or starting point are not specified
options(9)=-1; disp(errmes); disp(error1); return
end
if nargin<3, app=1; % No user-supplied gradients
elseif isempty(grad), app=1;
else, app=0; % Exact gradients are supplied
end
% OPTIONS ----{
doptions=[-1,1.e-4,1.e-6,15000,0,1.e-8,2.5,1e-11];
if nargin<4, options=doptions;
elseif isempty(options), options=doptions;
else,
% Replace default options by user specified options:
ii=find(options~=0);doptions(ii)=options(ii);
options=doptions;
end
% Check the values:
options([2:4,6:8])=abs(options([2:4,6:8]));
options(2:3)=max(options(2:3),[1.e-12,1.e-12]);
options(2)=max(options(8)*1.e2,options(2));
options(2:3)=min(options(2:3),[1,1]);
options(6)=max(options(6),1e-12);
options(7)=max([options(7),1.5]);
options(8)=max(options(8),1e-11);
% ----}
if nargin<5, constr=0; % Unconstrained problem
elseif isempty(func), constr=0;
else, constr=1; % Constrained problem
if nargin<6, appconstr=1; t=3; % No user-supplied gradients for constraints
elseif isempty(gradc),
appconstr=1;
else, appconstr=0; % Exact gradients of constraints are supplied
end
end
% ----}
% STARTING POINT ----{
if max(size(x))<=1, disp(errmes); disp(error2);
options(9)=-2; return
elseif size(x,2)==1, n=size(x,1); x=x'; trx=1;
elseif size(x,1)==1, n=size(x,2); trx=0;
else, disp(errmes); disp(error2);
options(9)=-2; return
end
% ----}
% WORKING CONSTANTS AND COUNTERS ----{
options(10)=0; options(11)=0; % function and gradient calculations
if constr
options(12)=0; options(13)=0; % same for constraints
end
epsnorm=1.e-15;epsnorm2=1.e-30; % epsilon & epsilon^2
if constr, h1=-1; % NLP: restricted to minimization
cnteps=options(6); % Max. admissible residual
else, h1=sign(options(1)); % Minimize resp. maximize a function
end
k=0; % Iteration counter
wdef=1/options(7)-1; % Default space transf. coeff.
%Gamma control ---{
ajb=1+.1/n^2; % Base I
ajp=20;
ajpp=ajp; % Start value for the power
ajs=1.15; % Base II
knorms=0; gnorms=zeros(1,10); % Gradient norms stored
%---}
%Display control ---{
if options(5)<=0, dispdata=0;
if options(5)==-1, dispwarn=0; else, dispwarn=1; end
else, dispdata=round(options(5)); dispwarn=1;
end, ld=dispdata;
%---}
%Stepsize control ---{
dq=5.1; % Step divider (at f_{i+1}>gamma*f_{i})
du20=2;du10=1.5;du03=1.05; % Step multipliers (at certain steps made)
kstore=3;nsteps=zeros(1,kstore); % Steps made at the last 'kstore' iterations
if app, des=6.3; % Desired number of steps per 1-D search
else, des=3.3; end
mxtc=3; % Number of trial cycles (steep wall detect)
%---}
termx=0; limxterm=50; % Counter and limit for x-criterion
ddx =max(1e-11,options(8)); % stepsize for gradient approximation
low_bound=-1+1e-4; % Lower bound cosine used to detect a ravine
ZeroGrad=n*1.e-16; % Lower bound for a gradient norm
nzero=0; % Zero-gradient events counter
% Lower bound for values of variables taking into account
lowxbound=max([options(2),1e-3]);
% Lower bound for function values to be considered as making difference
lowfbound=options(3)^2;
krerun=0; % Re-run events counter
detfr=options(3)*100; % relative error for f/f_{record}
detxr=options(2)*10; % relative error for norm(x)/norm(x_{record})
warnno=0; % the number of warn.mess. to end with
kflat=0; % counter for points of flatness
stepvanish=0; % counter for vanished steps
stopf=0;
% ----} End of setting constants
% ----} End of the preamble
% COMPUTE THE FUNCTION ( FIRST TIME ) ----{
if trx, f=feval(fun,x',pass);
else, f=feval(fun,x,pass); end
options(10)=options(10)+1;
if isempty(f), if dispwarn,disp(errmes);disp(error30);end
options(9)=-3; if trx, x=x';end, return
elseif isnan(f), if dispwarn,disp(errmes);disp(error31);disp(error6);end
options(9)=-3; if trx, x=x';end, return
elseif abs(f)==Inf, if dispwarn,disp(errmes);disp(error32);disp(error6);end
options(9)=-3; if trx, x=x';end, return
end
xrec=x; frec=f; % record point and function value
% Constrained problem
if constr, fp=f; kless=0;
if trx, fc=feval(func,x',pass);
else, fc=feval(func,x,pass); end
if isempty(fc),
if dispwarn,disp(errmes);disp(error50);end
options(9)=-5; if trx, x=x';end, return
elseif isnan(fc),
if dispwarn,disp(errmes);disp(error51);disp(error6);end
options(9)=-5; if trx, x=x';end, return
elseif abs(fc)==Inf,
if dispwarn,disp(errmes);disp(error52);disp(error6);end
options(9)=-5; if trx, x=x';end, return
end
options(12)=options(12)+1;
PenCoef=1; % first rough approximation
if fc<=cnteps, FP=1; fc=0; % feasible point
else, FP=0; % infeasible point
end
f=f+PenCoef*fc;
end
% ----}
% COMPUTE THE GRADIENT ( FIRST TIME ) ----{
if app, deltax=h1*ddx*ones(size(x));
if constr, if trx, g=apprgrdn2(x',fp,fun,pass,deltax',1);
else, g=apprgrdn2(x ,fp,fun,pass,deltax,1); end
else, if trx, g=apprgrdn2(x',f,fun,pass,deltax',1);
else, g=apprgrdn2(x ,f,fun,pass,deltax,1); end
end, options(10)=options(10)+n;
else, if trx, g=feval(grad,x',pass);
else, g=feval(grad,x,pass); end
options(11)=options(11)+1;
end
if size(g,2)==1, g=g'; end, ng=norm(g);
if size(g,2)~=n, if dispwarn,disp(errmes);disp(error40);end
options(9)=-4; if trx, x=x';end, return
elseif isnan(ng), if dispwarn,disp(errmes);disp(error41);disp(error6);end
options(9)=-4; if trx, x=x';end, return
elseif ng==Inf, if dispwarn,disp(errmes);disp(error42);disp(error6);end
options(9)=-4; if trx, x=x';end, return
elseif ng<ZeroGrad, if dispwarn,disp(errmes);disp(error43);disp(error6);end
options(9)=-4; if trx, x=x';end, return
end
if constr, if ~FP
if appconstr,
deltax=sign(x); idx=find(deltax==0);
deltax(idx)=ones(size(idx)); deltax=ddx*deltax;
if trx, gc=apprgrdn2(x',fc,func,pass,deltax',0);
else, gc=apprgrdn2(x ,fc,func,pass,deltax ,0); end
options(12)=options(12)+n;
else, if trx, gc=feval(gradc,x',pass);
else, gc=feval(gradc,x,pass); end
options(13)=options(13)+1;
end
if size(gc,2)==1, gc=gc'; end, ngc=norm(gc);
if size(gc,2)~=n,
if dispwarn,disp(errmes);disp(error60);end
options(9)=-6; if trx, x=x';end, return
elseif isnan(ngc),
if dispwarn,disp(errmes);disp(error61);disp(error6);end
options(9)=-6; if trx, x=x';end, return
elseif ngc==Inf,
if dispwarn,disp(errmes);disp(error62);disp(error6);end
options(9)=-6; if trx, x=x';end, return
elseif ngc<ZeroGrad,
if dispwarn,disp(errmes);disp(error63);end
options(9)=-6; if trx, x=x';end, return
end
g=g+PenCoef*gc; ng=norm(g);
end, end
grec=g; nng=ng;
% ----}
% INITIAL STEPSIZE
h=h1*sqrt(options(2))*max(abs(x)); % smallest possible stepsize
if abs(options(1))~=1,
h=h1*max(abs([options(1),h])); % user-supplied stepsize
else,
h=h1*max(1/log(ng+1.1),abs(h)); % calculated stepsize
end
% RESETTING LOOP ----{
while 1,
kcheck=0; % Set checkpoint counter.
kg=0; % stepsizes stored
kj=0; % ravine jump counter
B=eye(n); % re-set transf. matrix to identity
fst=f; g1=g; dx=0;
% ----}
% MAIN ITERATIONS ----{
while 1,
k=k+1;kcheck=kcheck+1;
laststep=dx;
% ADJUST GAMMA --{
gamma=1+max([ajb^((ajp-kcheck)*n),2*options(3)]);
gamma=min([gamma,ajs^max([1,log10(nng+1)])]);
% --}
gt=g*B; w=wdef;
% JUMPING OVER A RAVINE ----{
if (gt/norm(gt))*(g1'/norm(g1))<low_bound
if kj==2, xx=x; end, if kj==0, kd=4; end,
kj=kj+1; w=-.9; h=h*2; % use large coef. of space dilation
if kj>2*kd, kd=kd+1; warnno=1;
if any(abs(x-xx)<epsnorm*abs(x)), % flat bottom is detected
if dispwarn & ~shhh,disp(wrnmes);disp(warn08); end
end
end
else, kj=0;
end
% ----}
% DILATION ----{
z=gt-g1;
nrmz=norm(z);
if(nrmz>epsnorm*norm(gt))
z=z/nrmz;
g1=gt+w*(z*gt')*z; B=B+w*(B*z')*z;
else
z=zeros(1,n); nrmz=0; g1=gt;
end
d1=norm(g1); g0=(g1/d1)*B';
% ----}
% RESETTING ----{
if kcheck>1
idx=find(abs(g)>ZeroGrad); numelem=size(idx,2);
if numelem>0, grbnd=epsnorm*numelem^2;
if all(abs(g1(idx))<=abs(g(idx))*grbnd) | nrmz==0
if dispwarn & ~shhh, disp(wrnmes); disp(warn20); end
if abs(fst-f)<abs(f)*.01, ajp=ajp-10*n;
else, ajp=ajpp; end
h=h1*dx/3; k=k-1;
break
end
end
end
% ----}
% STORE THE CURRENT VALUES AND SET THE COUNTERS FOR 1-D SEARCH
xopt=x;fopt=f; k1=0;k2=0;ksm=0;kc=0;knan=0; hp=h;
if constr, Reset=0; end
% 1-D SEARCH ----{
while 1,
x1=x;f1=f;
if constr, FP1=FP; fp1=fp; end
x=x+hp*g0;
% FUNCTION VALUE
if trx, f=feval(fun,x',pass);
else, f=feval(fun,x,pass ); end
options(10)=options(10)+1;
if h1*f==Inf
if dispwarn, disp(errmes); disp(error5); end
options(9)=-7; if trx, x=x';end, return
end
if constr, fp=f;
if trx,fc=feval(func,x',pass);
else, fc=feval(func,x,pass);end
options(12)=options(12)+1;
if isnan(fc),
if dispwarn,disp(errmes);disp(error51);disp(error6);end
options(9)=-5; if trx, x=x';end, return
elseif abs(fc)==Inf,
if dispwarn,disp(errmes);disp(error52);disp(error6);end
options(9)=-5; if trx, x=x';end, return
end
if fc<=cnteps, FP=1; fc=0;
else, FP=0;
fp_rate=(fp-fp1);
if fp_rate<-epsnorm
if ~FP1
PenCoefNew=-15*fp_rate/norm(x-x1);
if PenCoefNew>1.2*PenCoef,
PenCoef=PenCoefNew; Reset=1; kless=0; f=f+PenCoef*fc; break
end
end
end
end
f=f+PenCoef*fc;
end
if abs(f)==Inf | isnan(f)
if dispwarn & ~shhh, disp(wrnmes);
if isnan(f), disp(error31); else, disp(error32); end
end
if ksm | kc>=mxtc, options(9)=-3; if trx, x=x';end, return
else, k2=k2+1;k1=0; hp=hp/dq; x=x1;f=f1; knan=1;
if constr, FP=FP1; fp=fp1; end
end
% STEP SIZE IS ZERO TO THE EXTENT OF EPSNORM
elseif all(abs(x-x1)<abs(x)*epsnorm),
stepvanish=stepvanish+1;
if stepvanish>=5,
options(9)=-14;
if dispwarn, disp(termwarn1);
disp(endwarn(4,:)); end
if trx,x=x';end, return
else, x=x1; f=f1; hp=hp*10; ksm=1;
if constr, FP=FP1; fp=fp1; end
end
% USE SMALLER STEP
elseif h1*f<h1*gamma^sign(f1)*f1
if ksm,break,end, k2=k2+1;k1=0; hp=hp/dq; x=x1;f=f1;
if constr, FP=FP1; fp=fp1; end
if kc>=mxtc, break, end
% 1-D OPTIMIZER IS LEFT BEHIND
else if h1*f<=h1*f1, break, end
% USE LARGER STEP
k1=k1+1; if k2>0, kc=kc+1; end, k2=0;
if k1>=20, hp=du20*hp;
elseif k1>=10, hp=du10*hp;
elseif k1>=3, hp=du03*hp;
end
end
end
% ----} End of 1-D search
% ADJUST THE TRIAL STEP SIZE ----{
dx=norm(xopt-x);
if kg<kstore, kg=kg+1; end
if kg>=2, nsteps(2:kg)=nsteps(1:kg-1); end
nsteps(1)=dx/(abs(h)*norm(g0));
kk=sum(nsteps(1:kg).*[kg:-1:1])/sum([kg:-1:1]);
if kk>des, if kg==1, h=h*(kk-des+1);
else, h=h*sqrt(kk-des+1); end
elseif kk<des, h=h*sqrt(kk/des);
end
stepvanish=stepvanish+ksm;
% ----}
% COMPUTE THE GRADIENT ----{
if app,
deltax=sign(g0); idx=find(deltax==0);
deltax(idx)=ones(size(idx)); deltax=h1*ddx*deltax;
if constr, if trx, g=apprgrdn2(x',fp,fun,pass,deltax',1);
else, g=apprgrdn2(x ,fp,fun,pass,deltax ,1); end
else, if trx, g=apprgrdn2(x',f,fun,pass,deltax',1);
else, g=apprgrdn2(x ,f,fun,pass,deltax ,1); end
end, options(10)=options(10)+n;
else
if trx, g=feval(grad,x',pass);
else, g=feval(grad,x ,pass); end
options(11)=options(11)+1;
end
if size(g,2)==1, g=g'; end, ng=norm(g);
if isnan(ng),
if dispwarn, disp(errmes); disp(error41); end
options(9)=-4; if trx, x=x'; end, return
elseif ng==Inf,
if dispwarn,disp(errmes);disp(error42);end
options(9)=-4; if trx, x=x';end, return
elseif ng<ZeroGrad,
if dispwarn & ~shhh,disp(wrnmes);disp(warn1);end
ng=ZeroGrad;
end
% Constraints:
if constr, if ~FP
if ng<.01*PenCoef
kless=kless+1;
if kless>=20, PenCoef=PenCoef/10; Reset=1; kless=0; end
else, kless=0;
end
if appconstr,
deltax=sign(x); idx=find(deltax==0);
deltax(idx)=ones(size(idx)); deltax=ddx*deltax;
if trx, gc=apprgrdn2(x',fc,func,pass,deltax',0);
else, gc=apprgrdn2(x ,fc,func,pass,deltax ,0); end
options(12)=options(12)+n;
else, if trx, gc=feval(gradc,x',pass);
else, gc=feval(gradc,x ,pass); end
options(13)=options(13)+1;
end
if size(gc,2)==1, gc=gc'; end, ngc=norm(gc);
if isnan(ngc),
if dispwarn,disp(errmes);disp(error61);end
options(9)=-6; if trx, x=x';end, return
elseif ngc==Inf,
if dispwarn,disp(errmes);disp(error62);end
options(9)=-6; if trx, x=x';end, return
elseif ngc<ZeroGrad & ~appconstr,
if dispwarn,disp(errmes);disp(error63);end
options(9)=-6; if trx, x=x';end, return
end
g=g+PenCoef*gc; ng=norm(g);
if Reset, if dispwarn & ~shhh, disp(wrnmes); disp(warn21); end
h=h1*dx/3; k=k-1; nng=ng; break
end
end, end
if h1*f>h1*frec, frec=f; xrec=x; grec=g; end
% ----}
if ng>ZeroGrad,
if knorms<10, knorms=knorms+1; end
if knorms>=2, gnorms(2:knorms)=gnorms(1:knorms-1); end
gnorms(1)=ng;
nng=(prod(gnorms(1:knorms)))^(1/knorms);
end
% DISPLAY THE CURRENT VALUES ----{
if k==ld
disp('Iter.# ..... Function ... Step Value ... Gradient Norm ');
disp(sprintf('%5i %13.5e %13.5e %13.5e',k,f,dx,ng));
ld=k+dispdata;
end
%----}
% CHECK THE STOPPING CRITERIA ----{
termflag=1;
if constr, if ~FP, termflag=0; end, end
if kcheck<=5, termflag=0; end
if knan, termflag=0; end
if kc>=mxtc, termflag=0; end
% ARGUMENT
if termflag
idx=find(abs(x)>=lowxbound);
if isempty(idx) | all(abs(xopt(idx)-x(idx))<=options(2)*abs(x(idx)))
termx=termx+1;
% FUNCTION
if abs(f-frec)> detfr * abs(f) & ...
abs(f-fopt)<=options(3)*abs(f) & ...
krerun<=3 & ...
~constr
if any(abs(xrec(idx)-x(idx))> detxr * abs(x(idx)))
if dispwarn & ~shhh,disp(wrnmes);disp(warn09);end
x=xrec; f=frec; g=grec; ng=norm(g); krerun=krerun+1;
h=h1*max([dx,detxr*norm(x)])/krerun;
warnno=2; break
else, h=h*10;
end
elseif abs(f-frec)> options(3)*abs(f) & ...
norm(x-xrec)<options(2)*norm(x) & constr
elseif abs(f-fopt)<=options(3)*abs(f) | ...
abs(f)<=lowfbound | ...
(abs(f-fopt)<=options(3) & termx>=limxterm )
if stopf
if dx<=laststep
if warnno==1 & ng<sqrt(options(3)), warnno=0; end
if ~app, if any(abs(g)<=epsnorm2), warnno=3; end, end
if warnno~=0, options(9)=-warnno-10;
if dispwarn, disp(termwarn1);
disp(endwarn(warnno,:));
if app, disp(appwarn); end
end
else, options(9)=k;
if dispwarn, disp(termwarn0); end
end
if trx,x=x';end, return
end
else, stopf=1;
end
elseif dx<1.e-12*max(norm(x),1) & termx>=limxterm
options(9)=-14;
if dispwarn, disp(termwarn1); disp(endwarn(4,:));
if app, disp(appwarn); end
end
x=xrec; f=frec;
if trx,x=x';end, return
else, stopf=0;
end
end
end
% ITERATIONS LIMIT
if(k==options(4))
options(9)=-9; if trx, x=x'; end,
if dispwarn & ~shhh, disp(wrnmes); disp(warn4); end
return
end
% ----}
% ZERO GRADIENT ----{
if constr
if ng<=ZeroGrad,
if dispwarn, disp(termwarn1); disp(warn1); end
options(9)=-8; if trx,x=x';end,return
end
else
if ng<=ZeroGrad, nzero=nzero+1;
if dispwarn & ~shhh, disp(wrnmes); disp(warn1); end
if nzero>=3, options(9)=-8; if trx,x=x';end,return, end
g0=-h*g0/2;
for i=1:10,
x=x+g0;
if trx, f=feval(fun,x',pass);
else, f=feval(fun,x ,pass); end
options(10)=options(10)+1;
if abs(f)==Inf
if dispwarn, disp(errmes); disp(error32); end
options(9)=-3;if trx,x=x';end,return
elseif isnan(f),
if dispwarn, disp(errmes); disp(error32); end
options(9)=-3;if trx,x=x';end,return
end
if app,
deltax=sign(g0); idx=find(deltax==0);
deltax(idx)=ones(size(idx)); deltax=h1*ddx*deltax;
if trx, g=apprgrdn2(x',f,fun,pass,deltax',1);
else, g=apprgrdn2(x ,f,fun,pass,deltax ,1); end
options(10)=options(10)+n;
else
if trx, g=feval(grad,x',pass);
else, g=feval(grad,x ,pass); end
options(11)=options(11)+1;
end
if size(g,2)==1, g=g'; end, ng=norm(g);
if ng==Inf
if dispwarn, disp(errmes); disp(error42); end
options(9)=-4; if trx, x=x'; end, return
elseif isnan(ng)
if dispwarn, disp(errmes); disp(error41); end
options(9)=-4; if trx, x=x'; end, return
end
if ng>ZeroGrad, break, end
end
if ng<=ZeroGrad,
if dispwarn, disp(termwarn1); disp(warn1); end
options(9)=-8; if trx,x=x';end,return
end
h=h1*dx; break
end
end
% ----}
% FUNCTION IS FLAT AT THE POINT ----{
if ~constr & abs(f-fopt)<abs(fopt)*options(3) & kcheck>5 & ng<1
idx=find(abs(g)<=epsnorm2); ni=size(idx,2);
if ni>=1 & ni<=n/2 & kflat<=3, kflat=kflat+1;
if dispwarn & ~shhh, disp(wrnmes); disp(warn31); end, warnno=1;
x1=x; fm=f;
for j=idx, y=x(j); f2=fm;
if y==0, x1(j)=1; elseif abs(y)<1, x1(j)=sign(y); else, x1(j)=y; end
for i=1:20, x1(j)=x1(j)/1.15;
if trx, f1=feval(fun,x1',pass);
else, f1=feval(fun,x1 ,pass); end
options(10)=options(10)+1;
if abs(f1)~=Inf & ~isnan(f1),
if h1*f1>h1*fm, y=x1(j); fm=f1;
elseif h1*f2>h1*f1, break
elseif f2==f1, x1(j)=x1(j)/1.5;
end, f2=f1;
end
end
x1(j)=y;
end
if h1*fm>h1*f
if app,
deltax=h1*ddx*ones(size(deltax));
if trx, gt=apprgrdn2(x1',fm,fun,pass,deltax',1);
else, gt=apprgrdn2(x1 ,fm,fun,pass,deltax ,1); end
options(10)=options(10)+n;
else
if trx, gt=feval(grad,x1',pass);
else, gt=feval(grad,x1 ,pass); end
options(11)=options(11)+1;
end
if size(gt,2)==1, gt=gt'; end, ngt=norm(gt);
if ~isnan(ngt) & ngt>epsnorm2,
if dispwarn, disp(warn32); end
options(3)=options(3)/5;
x=x1; g=gt; ng=ngt; f=fm; h=h1*dx/3; break
end
end
end
end
% ----}
end % iterations
end % restart
% end of the function
%
function g = apprgrdn2(x,f,fun,pass,deltax,obj)
% added 'pass' input to pass params to called function
% Usage:
% g = apprgrdn(x,f,fun,deltax,obj)
% Function apprgrdn.m performs the finite difference approximation
% of the gradient <g> at a point <x>.
% <f> is the calculated function value at a point <x>,
% <fun> is the name of the Matlab function, which calculates function values
% <deltax> is a vector of the relative stepsizes,
% <obj> is the flag indicating whether the gradient of the objective
% function (1) or the constraint function (0) is to be calculated.
%
n=max(size(x)); ee=ones(size(x));
di=abs(x); idx=find(di<5e-15); di(idx)=5e-15*ee(idx);
di=deltax.*di;
if obj, idx=find(abs(di)<2e-10); di(idx)=2e-10*sign(di(idx));
else, idx=find(abs(di)<5e-15); di(idx)=5e-15*sign(di(idx));
end
y=x;
for i=1:n
y(i)=x(i)+di(i);
fi=feval(fun,y,pass);
if obj, if fi==f,
for j=1:3
di(i)=di(i)*10; y(i)=x(i)+di(i);
fi=feval(fun,y,pass); if fi~=f, break, end
end
end, end
g(i)=(fi-f)/di(i);
if obj, if any(idx==i)
y(i)=x(i)-di(i);
fi=feval(fun,y,pass);
g(i)=.5*(g(i)+(f-fi)/di(i));
end, end
y(i)=x(i);
end
